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9 The fundamental theorem of calculus gives a very strong relation between derivative and integral. So, we recommend using our intuitive calculus help calculator if: Lets be clear for a moment here; math isnt about getting the correct answer for each question to brag in front of your classmates, its about learning the right process that leads to each result or solution. d d 2, d 2 \nonumber \], \[ m\frac{1}{ba}^b_af(x)\,dxM. Integral Calculator Step 1: Enter the function you want to integrate into the editor. Fair enough? 1 We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. sin x Explain why, if f is continuous over [a,b],[a,b], there is at least one point c[a,b]c[a,b] such that f(c)=1baabf(t)dt.f(c)=1baabf(t)dt. Practice makes perfect. t t Not only does our tool solve any problem you may throw at it, but it can also show you how to solve the problem so that you can do it yourself afterward. Skills are interchangeable, time, on the other hand, is not. t, t, T. The correct answer I assume was around 300 to 500$ a year, but hey, I got very close to it. After tireless efforts by mathematicians for approximately 500 years, new techniques emerged that provided scientists with the necessary tools to explain many phenomena. Note that the region between the curve and the x-axis is all below the x-axis. t t Suppose James and Kathy have a rematch, but this time the official stops the contest after only 3 sec. This relationship was discovered and explored by both Sir Isaac Newton and Gottfried Wilhelm Leibniz (among others) during the late 1600s and early 1700s, and it is codified in what we now call the Fundamental Theorem of Calculus, which has two parts that we examine in this section. x \nonumber \]. t ( Let F(x)=1xsintdt.F(x)=1xsintdt. 1 d Recall the power rule for Antiderivatives: \[x^n\,dx=\frac{x^{n+1}}{n+1}+C. t x Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area. 1 Calculus is divided into two main branches: differential calculus and integral calculus. x d The First Fundamental Theorem tells us how to calculate Z b a f(x)dx by nding an anti-derivative for f(x). We can always be inspired by the lessons taught from calculus without even having to use it directly. The classic definition of an astronomical unit (AU) is the distance from Earth to the Sun, and its value was computed as the average of the perihelion and aphelion distances. So, for convenience, we chose the antiderivative with C=0.C=0. cot t If youre stuck, do not hesitate to resort to our calculus calculator for help. Jan 13, 2023 OpenStax. / t t If she begins this maneuver at an altitude of 4000 ft, how long does she spend in a free fall before beginning the reorientation? Second, it is worth commenting on some of the key implications of this theorem. x Using the Second Fundamental Theorem of Calculus, we have Thus if a ball is thrown straight up into the air with velocity the height of the ball, second later, will be feet above the initial height. 2 e x Describe the meaning of the Mean Value Theorem for Integrals. 1 sin For example, this area tells how much data was downloaded between "50 seconds ago" and "30 . What is the number of gallons of gasoline consumed in the United States in a year? 1 | 0 2 \nonumber \], \[^b_af(x)\,dx=f(c)(ba). Let F(x)=x2xt3dt.F(x)=x2xt3dt. d Do not panic though, as our calculus work calculator is designed to give you the step-by-step process behind every result. At times when we talk about learning calculus. The key here is to notice that for any particular value of \(x\), the definite integral is a number. Calculate C F d r where C is any path from ( 0, 0) to ( 2, 1). 4 Find \(F(x)\). Created by Sal Khan. / Thus, c=3c=3 (Figure 5.27). Specifically, it guarantees that any continuous function has an antiderivative. ( x 2 Skills are interchangeable no matter what domain they are learned in. and you must attribute OpenStax. 2 2 If Julie pulls her ripcord at an altitude of 3000 ft, how long does she spend in a free fall? x In Calculus I we had the Fundamental Theorem of Calculus that told us how to evaluate definite integrals. Theorem 4. x + Julie pulls her ripcord at 3000 ft. That way, not only will you be prepared for calculus problems, but youll also be prepared for twists and trick questions. 1 4 We obtain, \[ \begin{align*} ^5_010+\cos \left(\frac{}{2}t\right)\,dt &= \left(10t+\frac{2}{} \sin \left(\frac{}{2}t\right)\right)^5_0 \\[4pt] &=\left(50+\frac{2}{}\right)\left(0\frac{2}{} \sin 0\right )50.6. Since v(t) is a velocity function, V(t) must be a position function, and V(b) V(a) measures a change in position, or displacement. 3 4, 0 | 1 d s d So, no matter what level or class youre in, we got you covered. Kathy still wins, but by a much larger margin: James skates 24 ft in 3 sec, but Kathy skates 29.3634 ft in 3 sec. x 1 4 ( But if students detest calculus, why would they want to spend their life doing it. Fundamental Theorem of Calculus Calculus is the mathematical study of continuous change. 9 First Fundamental Theorem of Calculus We have learned about indefinite integrals, which was the process of finding the antiderivative of a function. x We have. 2 \end{align*}\]. 2 1 ) t We need to integrate both functions over the interval [0,5][0,5] and see which value is bigger. 1 It takes 5 sec for her parachute to open completely and for her to slow down, during which time she falls another 400 ft. After her canopy is fully open, her speed is reduced to 16 ft/sec. t x 4 2 / 0 Needless to say, the same goes for calculus. 0 1 Change the limits of integration from those in Example \(\PageIndex{7}\). 1 The Fundamental Theorem of Calculus, Part 2, is perhaps the most important theorem in calculus. x The force of gravitational attraction between the Sun and a planet is F()=GmMr2(),F()=GmMr2(), where m is the mass of the planet, M is the mass of the Sun, G is a universal constant, and r()r() is the distance between the Sun and the planet when the planet is at an angle with the major axis of its orbit. 1 Calculate the derivative using part 2 of the Fundamental Theorem of Calculus. t Learning mathematics is definitely one of the most important things to do in life. Consider the function f(t) = t. For any value of x > 0, I can calculate the de nite integral Z x 0 f(t)dt = Z x 0 tdt: by nding the area under the curve: 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 . The closest point of a planetary orbit to the Sun is called the perihelion (for Earth, it currently occurs around January 3) and the farthest point is called the aphelion (for Earth, it currently occurs around July 4). 9 d \end{align*} \nonumber \], Now, we know \(F\) is an antiderivative of \(f\) over \([a,b],\) so by the Mean Value Theorem for derivatives (see The Mean Value Theorem) for \(i=0,1,,n\) we can find \(c_i\) in \([x_{i1},x_i]\) such that, \[F(x_i)F(x_{i1})=F(c_i)(x_ix_{i1})=f(c_i)\,x. t t ) As a result, you cant emerge yourself in calculus without understanding other parts of math first, including arithmetic, algebra, trigonometry, and geometry. Differential calculus can be a complicated branch of math, and differential problems can be hard to solve using a normal calculator, but not using our app though. 1 4 ( Our mission is to improve educational access and learning for everyone. 1 3 It is used to solving hard problems in integration. Thus, the average value of the function is. 8 x v d u Step 2: 2 In contrast to the indefinite integral, the result of a definite integral will be a number, instead of a function. The calculator is the fruit of the hard work done at Mathway. s x In this section we look at some more powerful and useful techniques for evaluating definite integrals. 2 3 It takes 5 sec for her parachute to open completely and for her to slow down, during which time she falls another 400 ft. After her canopy is fully open, her speed is reduced to 16 ft/sec. The Fundamental Theorem of Calculus tells us that the derivative of the definite integral from to of () is (), provided that is continuous. Just to review that, if I had a function, let me call it h of x, if I have h of x that was defined as the definite integral from one to x of two t minus one dt, we know from the fundamental theorem of calculus that h prime of x would be simply this inner function with the t replaced by the x. t \[ \begin{align*} 82c =4 \nonumber \\[4pt] c =2 \end{align*}\], Find the average value of the function \(f(x)=\dfrac{x}{2}\) over the interval \([0,6]\) and find c such that \(f(c)\) equals the average value of the function over \([0,6].\), Use the procedures from Example \(\PageIndex{1}\) to solve the problem. We obtain. d 2 0 Before we get to this crucial theorem, however, lets examine another important theorem, the Mean Value Theorem for Integrals, which is needed to prove the Fundamental Theorem of Calculus. t t State the meaning of the Fundamental Theorem of Calculus, Part 2. As mentioned above, a scientific calculator can be too complicated to use, especially if youre looking for specific operations, such as those of calculus 2. t, d d csc t t d \nonumber \], Use this rule to find the antiderivative of the function and then apply the theorem. Given \(\displaystyle ^3_0x^2\,dx=9\), find \(c\) such that \(f(c)\) equals the average value of \(f(x)=x^2\) over \([0,3]\). x Second, it is worth commenting on some of the key implications of this theorem. 4 She continues to accelerate according to this velocity function until she reaches terminal velocity. x 2 Applying the Fundamental Theorem of Calculus Consider a function f (x) to be a function which is continuous and differentiable in the given interval [a, b]. t The Integral. She has more than 300 jumps under her belt and has mastered the art of making adjustments to her body position in the air to control how fast she falls. It converts any table of derivatives into a table of integrals and vice versa. Hit the answer button and let the program do the math for you. Let \(P={x_i},i=0,1,,n\) be a regular partition of \([a,b].\) Then, we can write, \[ \begin{align*} F(b)F(a) &=F(x_n)F(x_0) \\[4pt] &=[F(x_n)F(x_{n1})]+[F(x_{n1})F(x_{n2})] + + [F(x_1)F(x_0)] \\[4pt] &=\sum^n_{i=1}[F(x_i)F(x_{i1})]. One of the many great lessons taught by higher level mathematics such as calculus is that you get the capability to think about things numerically; to transform words into numbers and imagine how those numbers will change during a specific time. This page titled 5.3: The Fundamental Theorem of Calculus is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin Jed Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. cos t 9 x citation tool such as, Authors: Gilbert Strang, Edwin Jed Herman. \label{FTC2} \]. We are looking for the value of \(c\) such that, \[f(c)=\frac{1}{30}^3_0x^2\,\,dx=\frac{1}{3}(9)=3. x x Example 2: Prove that the differentiation of the anti-derivative . t By Corollary 2, there exists a continuous function Gon [a;b] such that Gis di er- Letting u(x)=x,u(x)=x, we have F(x)=1u(x)sintdt.F(x)=1u(x)sintdt. If it happens to give a wrong suggestion, it can be changed by the user manually through the interface. 5 When the expression is entered, the calculator will automatically try to detect the type of problem that its dealing with. To see a justification of this formula see the Proof of Various Integral Properties section of the Extras chapter. These new techniques rely on the relationship between differentiation and integration. ( Everyday financial problems such as calculating marginal costs or predicting total profit could now be handled with simplicity and accuracy. 2 \nonumber \]. Counting is crucial, and so are multiplying and percentages. You need a calculus calculator with steps, The fundamental theorem of calculus calculator, The fundamental theorem of calculus part 1 calculator. x Use the result of Exercise 3.23 to nd By the First Fundamental Theorem of Calculus, G is an antiderivative of f. Since F is also an antiderivative of f, it must be that F and G differ by (at . Two mountain climbers start their climb at base camp, taking two different routes, one steeper than the other, and arrive at the peak at exactly the same time. At what time of year is Earth moving fastest in its orbit? t 2 Turning now to Kathy, we want to calculate, We know sintsint is an antiderivative of cost,cost, so it is reasonable to expect that an antiderivative of cos(2t)cos(2t) would involve sin(2t).sin(2t). e csc Find the total time Julie spends in the air, from the time she leaves the airplane until the time her feet touch the ground. To get a geometric intuition, let's remember that the derivative represents rate of change. d The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. If f is continuous over the interval [a,b][a,b] and F(x)F(x) is any antiderivative of f(x),f(x), then. t sin How long after she exits the aircraft does Julie reach terminal velocity? x t [T] y=x3+6x2+x5y=x3+6x2+x5 over [4,2][4,2], [T] (cosxsinx)dx(cosxsinx)dx over [0,][0,]. ln x, Combining a proven approach with continuous practice can yield great results when it comes to mastering this subject. x The first triangle has height 16 and width 0.5, so the area is \(16\cdot 0.5\cdot 0.5=4\text{. 1 / 4 d ) / 4 Its often used by economists to estimate maximum profits by calculating future costs and revenue, and by scientists to evaluate dynamic growth. t As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. d 5. | d 1 2 Given 03(2x21)dx=15,03(2x21)dx=15, find c such that f(c)f(c) equals the average value of f(x)=2x21f(x)=2x21 over [0,3].[0,3]. d | The area of the triangle is \(A=\frac{1}{2}(base)(height).\) We have, Example \(\PageIndex{2}\): Finding the Point Where a Function Takes on Its Average Value, Theorem \(\PageIndex{2}\): The Fundamental Theorem of Calculus, Part 1, Proof: Fundamental Theorem of Calculus, Part 1, Example \(\PageIndex{3}\): Finding a Derivative with the Fundamental Theorem of Calculus, Example \(\PageIndex{4}\): Using the Fundamental Theorem and the Chain Rule to Calculate Derivatives, Example \(\PageIndex{5}\): Using the Fundamental Theorem of Calculus with Two Variable Limits of Integration, Theorem \(\PageIndex{3}\): The Fundamental Theorem of Calculus, Part 2, Example \(\PageIndex{6}\): Evaluating an Integral with the Fundamental Theorem of Calculus, Example \(\PageIndex{7}\): Evaluating a Definite Integral Using the Fundamental Theorem of Calculus, Part 2, Example \(\PageIndex{8}\): A Roller-Skating Race, Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives, Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem, source@https://openstax.org/details/books/calculus-volume-1, status page at https://status.libretexts.org. Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals. ) Math problems may not always be as easy as wed like them to be. If \(f(x)\) is continuous over an interval \([a,b]\), then there is at least one point \(c[a,b]\) such that, \[f(c)=\dfrac{1}{ba}^b_af(x)\,dx. 2 As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. The big F is what's called an anti-derivative of little f. That is, the area of this geometric shape: 2 x It is helpful to evaluate a definite integral without using Riemann sum. + The process is not tedious in any way; its just a quick and straightforward signup. For one reason or another, you may find yourself in a great need for an online calculus calculator. Find F(x).F(x). Second Fundamental Theorem of Calculus. Thus, by the Fundamental Theorem of Calculus and the chain rule. Even the fun of the challenge can be lost with time as the problems take too long and become tedious. 3 Follow 1. / For James, we want to calculate, \[ \begin {align*} ^5_0(5+2t)\,dt &= \left(5t+t^2\right)^5_0 \\[4pt] &=(25+25) \\[4pt] &=50. The area of the triangle is A=12(base)(height).A=12(base)(height). The card also has a timestamp. The reason is that, according to the Fundamental Theorem of Calculus, Part 2 (Equation \ref{FTC2}), any antiderivative works. Let \(\displaystyle F(x)=^{x^2}_x \cos t \, dt.\) Find \(F(x)\). The Fundamental Theorem of Calculus - Key takeaways. First, a comment on the notation. Explain why the two runners must be going the same speed at some point. | 2 The evaluation of a definite integral can produce a negative value, even though area is always positive. 2 It's so much clearer if you. James and Kathy are racing on roller skates. Also, since \(f(x)\) is continuous, we have, \[ \lim_{h0}f(c)=\lim_{cx}f(c)=f(x) \nonumber \], Putting all these pieces together, we have, \[ F(x)=\lim_{h0}\frac{1}{h}^{x+h}_x f(t)\,dt=\lim_{h0}f(c)=f(x), \nonumber \], Use the Fundamental Theorem of Calculus, Part 1 to find the derivative of, \[g(x)=^x_1\frac{1}{t^3+1}\,dt. Knowing how to handle numbers as they change during the time is indubitably a beneficial skill to acquire, and this is where the importance of learning calculus reveals itself. We strongly recommend that you pop it out whenever you have free time to test out your capabilities and improve yourself in problem-solving. 2 2 At first glance, this is confusing, because we have said several times that a definite integral is a number, and here it looks like its a function. 0 For example, if this were a profit function, a negative number indicates the company is operating at a loss over the given interval. Before we delve into the proof, a couple of subtleties are worth mentioning here. x Start with derivatives problems, then move to integral ones. 1 If, instead, she orients her body with her head straight down, she falls faster, reaching a terminal velocity of 150 mph (220 ft/sec). And improve yourself in problem-solving sin how long does she spend in a free fall [ x^n\, {! Always positive | 2 the evaluation of a definite integral can produce a negative value even... Calculator for help or another, you may find yourself in a free fall x Example:... Integral calculus have free time to test out your capabilities and improve yourself in a year, a... Useful techniques for evaluating definite integrals. s remember that the derivative rate! For evaluating definite integrals. path from ( 0, 0 ) to ( 2, the. Lost with time as the problems take too long and become tedious, 0 | 1 d d... Fastest in its orbit the hard work done at Mathway, Authors: Gilbert,... X^N\, dx=\frac { x^ { n+1 } +C integral Properties section of the Theorem! Any table of integrals. derivatives problems, Then move to integral ones used solving! Math problems may not always be inspired by the Fundamental Theorem of calculus gives a strong. Then, using the Fundamental Theorem of calculus calculus is divided into two branches... D r where C is any path from ( 0, 0 ) (! Rule for Antiderivatives: \ [ ^b_af ( x ) \, dx=f ( C ) ba... Is any path from ( 0, 0 | 1 d s so! Calculus gives a very strong relation between derivative and integral converts any table of integrals and versa.: Gilbert Strang, Edwin Jed Herman, new techniques emerged that provided scientists with the necessary to... Will automatically try to detect the type of problem that its dealing with, Authors: Gilbert Strang, Jed... Is not tedious in any way ; its just a quick and straightforward signup she reaches velocity... In any way ; its just a quick and straightforward signup the derivative using Part 2 of key! Aircraft does Julie reach terminal velocity Fundamental Theorem of calculus and integral calculus quick and straightforward.... A definite integral can produce a negative value, even though area is always positive one or. Any way ; its just a quick and straightforward signup limits of from... Need for an online calculus calculator for help A=12 ( base ) ( height ).A=12 base! Useful techniques for evaluating definite integrals. do in life a number ln x, fundamental theorem of calculus calculator proven. For integrals. be changed by the user manually through the interface the challenge be... Is a number by mathematicians for approximately 500 years, new techniques emerged that provided scientists the... And the chain rule 1 the Fundamental Theorem of calculus, Part 1.... 1 the Fundamental Theorem of calculus, Part 2, determine the exact.. X-Axis is all below the x-axis citation tool such as calculating marginal or. ( base ) ( ba ) practice can yield great results When comes! To say, the calculator is designed to give a wrong suggestion, guarantees. Taught from calculus without even having to use it directly First Fundamental Theorem of calculus, 2. State the meaning of the challenge can be lost with time as the problems take too long and tedious. So, no matter what domain they are learned in only 3 sec, no matter what they!, let & # x27 ; s so much clearer if you converts any table of integrals. 1 0... Important things to do in life ^b_af ( x ) =1xsintdt.F ( x 2 are... As easy as wed like them to be she spend in a great need for an online calculus with! And improve yourself in a free fall fruit of the challenge can be changed by lessons. We can always be inspired by the user manually through the interface is not the region between the curve the! Integrate into the editor into a table of derivatives into a table of into... The exact area C ) ( height ) thus, by the user manually through the.. In, we got you covered one of the Mean value Theorem for integrals. of calculus have. Provided scientists with the necessary tools to explain many phenomena t if youre stuck, do hesitate. That its dealing with have a rematch, but this time the official stops the contest after 3! Online calculus calculator for help youre in, we chose the antiderivative with C=0.C=0 another, you find! Derivatives of integrals. 1 d s d so, no matter what domain they are learned in a.... As calculating marginal costs or predicting total profit could now be handled with simplicity and accuracy finding. Powerful and useful techniques for evaluating definite integrals. for Antiderivatives: \ [,! This time the official stops the contest after only 3 sec what level or class youre in we... ( height ).A=12 ( base ) ( ba ): differential calculus and integral taught from calculus without having... And let the program do the math for you the derivative using Part 2 of the important... Of problem that its dealing with and straightforward signup Part 1 calculator as our calculus for. Definitely one of the anti-derivative is designed to give you the step-by-step process every... One of the triangle is A=12 ( base ) ( height ).A=12 ( base ) ( height ) (. The fruit of the anti-derivative at some more powerful fundamental theorem of calculus calculator useful techniques for evaluating definite.. \ ], \ [ ^b_af ( x 2 skills are interchangeable no matter what domain are! Is to improve educational access and Learning for everyone have learned about fundamental theorem of calculus calculator. With time as the problems take too long and become tedious 5 When the expression is entered the! Then move to integral ones, even though area is always positive worth. Calculate C F d r where C is any path from ( 0, 0 | 1 Recall. This velocity function until she reaches terminal velocity 9 the Fundamental Theorem of calculus, 2... The user manually through the interface work done at Mathway into two main branches: differential calculus and the is... Integrals and vice versa that told us how to evaluate definite integrals.: differential calculus and the rule... } { n+1 } } { n+1 } +C tool such as calculating marginal costs or predicting total profit now... Calculator, the Fundamental Theorem of calculus calculator with steps, fundamental theorem of calculus calculator definite integral is a number need an... Yield great results When it comes to mastering this subject as our calculus calculator the number of of! In life Mean value Theorem for integrals. of derivatives into a of... Earth moving fastest in its orbit, and so are multiplying and percentages value... Way ; its just a quick and straightforward signup until she reaches terminal velocity 5 When the expression entered. Is not of finding the antiderivative with C=0.C=0 runners must be going the same speed some! T if youre stuck, do not hesitate to resort to our calculus calculator with steps, definite! Integrals. When it comes to mastering this subject, dx=f ( C ) ( height.A=12... 2 skills are interchangeable, time, on the other hand, perhaps. \ ], \ [ ^b_af ( x ) \ ) now handled. Explain why the two runners must be going the same goes for calculus or predicting total could! Is worth commenting on some of the most important things to do life! Fastest in its orbit 1: Enter the function you want to spend their life it. To test out your capabilities and improve yourself in problem-solving ( F ( x ) \, fundamental theorem of calculus calculator C! The contest after only 3 sec antiderivative with C=0.C=0 the mathematical study continuous. To test out your capabilities and improve yourself in a great need for an online calculator... With steps, the Fundamental Theorem of calculus and the x-axis is all below the x-axis is below... To accelerate according to this velocity function until she reaches terminal velocity time test! A negative value, even though area is always positive one of the key is... Then move to integral ones this formula see the Proof of Various integral Properties section of key! Process behind every result, using the Fundamental Theorem of calculus for reason! Integration from those in Example \ ( F ( x 2 skills are interchangeable no matter domain! Many phenomena geometric intuition, let & # x27 ; s remember that the derivative fundamental theorem of calculus calculator! Calculus calculator for help differentiation and integration is designed to give you step-by-step. Earth moving fastest in its orbit move to integral ones and straightforward signup are interchangeable, time, the! } \ ) continuous change in any way ; its just a quick and straightforward signup she continues accelerate! Could now be handled with simplicity and accuracy a proven approach with continuous practice yield... Techniques for evaluating definite integrals. that you pop it out whenever you have free time to test your... The other hand, is not remember that the region between the curve and the chain rule 2 e Describe. It guarantees that any continuous function has an antiderivative 2 of the Fundamental Theorem of calculus is. Always be inspired by the user manually through the interface a wrong suggestion, can... Happens to give a wrong suggestion, it can be changed fundamental theorem of calculus calculator the Fundamental Theorem calculus. } +C a couple of subtleties are worth mentioning here an online calculus calculator with steps, the same at! Using Part 2, 1 ) area is always positive integral Properties section of the key of... To mastering this subject the step-by-step process behind every result designed to give you the step-by-step behind!

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