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Calculus: The Area between Two Curves


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About this Lesson

  • Type: Video Tutorial
  • Length: 9:04
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 98 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Applications of Integration (10 lessons, $16.83)
Calculus: Finding Area between Two or More Curves (4 lessons, $7.92)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Applications of Integration
Finding the Area Between Two Curves
The Area between Two Curves Page [1 of 1]
Okay, well, welcome back. We just have been taking a look at some of those definite integral problems. I now want to return to the issue of a definite integrals representing area. What we've seen so far is that we want to find the area under a particular function, but above the x-axis. What we do is we just integrate the function and evaluate the integral that anti-derivative at B and subtract off what we get at A.
Okay, we've seen a lot of examples of those, and I hope you are getting a little more comfortable with that. Now, where does that come from? Well, it came from the basic notion of summing up rectangles, and if you put in, sort of, an arbitrary rectangle here so the rectangle has height, f(x), and the base and the tiny change in x. So, it will be base times height. We can take that very basic idea and now extend it. Instead of looking at areas under a curve and above the x-axis, how about the following? How about under a curve and above another curve? How about a really exotic region like something like this? Like maybe the region between these two curves. So, let's suppose, let me put in some axes here. I put in some axes; it doesn't make a difference where these are. Suppose I give you now--and one endpoint, let's say, A is way over here, call this A, and I call this point over here B. So, I go up and you see this. You go up here like that. What you see is I've now actually plotted out a really exotic region, the area that sits below the blue curve, but above the red curve. Look at that, that's a really exotic region. How would you find the area of that region?
Well, there's two ways of thinking about this problem. One way, is through the following: first, find the area under the blue, that's just the integral, that's pretty easy, and then to subtract off this little area here, so, now find the area under the red, and take the first answer and subtract off the second answer. That would give you the area of the whole thing, and then minus this piece here would give you the area that you want. That's one way of thinking about it. Another way of thinking about it is just to say, "Well, what are we summing up here?" So, that area would equal and I'm going to sum up, well still from A to B, so, from A to B, and what am I summing up? Well, I'm summing up these rectangles, and let me draw a generic rectangle in, and I love it if you get into the habit of drawing in just the generic rectangle to help you. So, understand that there are a lot of rectangles here, this is just a generic one I plucked out of the air. What's the area of that? Well, it's base, well, that's a tiny change in x, times the height. And what's the height? Well, the height is actually this value, but then subtracted off by this value. So, if this function would say, we're f(x) here, and this function were g(x) here, then I would see I would have of f(x) minus g(x). That's the height of this rectangle. And so actually, if you work this out, that integral would give you the area between two curves. And you can see, the other method there, well, first, is the integral from A to B is f(x)dx. So, this piece right here gives you the area under the blue curve all the way to here, and this piece right here gives you the area under the curve just under the red curve from here to here. So, if I take the entire answer and subtract off this little region right in here, then I get the yellow. So, in fact, finding areas between two curves is actually pretty easy, the important thing to do is to make sure you take the top function, the one that's on top, and then subtract off the one that's one the bottom. Always take top minus bottom and then you can find areas between two curves. So, that's pretty easy, let's try an example.
Look, Ma, no functions. So, how about find the area between the following curves. f(x) = x^2 + 4 and g(x) = x - 2, and I want you to find the area between these two curves between two points and the two points that I want you to find the areas are -1 and x=3. So, how do you proceed? Well, the first thing I think we should do is draw a sketch of this. We need to know which function's on top and which function is on the bottom. So, let's try to draw a little sketch of this picture, put these graphs together. Well, the first function f is a parabola, it's a good old fashioned standard parabola, happy faced parabola, should put up four units because we add the four. So, in fact that would look something like this. Roughly speaking from this point of the four, so this is f(x) here, that's a happy face parabola. What's this? Well, this is a line of slope 1, y intercept -2, so it's actually down here, -2, but the slope is 1. It looks like one of the standard lines that shift down a little bit. And so this is g(x). And where am I looking this region between? I'm looking at this region between -1, which is way over here, and 3; so one, two, three. So, you see the region that these two curves bound? Sort of right in between here, it's this region right in here. You want to find the area of all this stuff in between these two curves between -1 and 3.
Okay, well how do we set this up? Well, we just take our time and right the integral out and think about what exactly it means. So, the area would equal the integral. And where am I going to sum up these rectangles from? Well, from the very left out to the very right. So, from -1 all the way out to 3, and then what am I doing? I'm summing up the areas of the rectangles, so you might want to actually draw one in. This is what I would do and it's base times height. And what's the base? The base is a tiny change in x, so that's the dx. And what's the height? Well, the height is going to be the top function minus the bottom function. So, it's going to be a top function x^2 + 4 minus the bottom function, and the bottom function is x - 2. So, it's just the top minus the bottom, notice the parenthesis around that, because I have to subtract the whole thing. And, now what does that equal? Well, that equals the integral from -1 to 3 of x^2 and this 4- (-2) is a 6, and that is a - x, so -x + 6 dx, and you integrate that and I see x^3/3 - x^2/2 + 6x and I evaluate this from -1 up to 3. So, first I plug in 3 everywhere and then I plug in -1 everywhere. So, if I do that, what do I see? Let me do that little side calculations here where I copy the function. So, if I try that, what do I get? Well, what I get is I plug in 3 everywhere I see 3^3,^ which is 27/2 minus 3^2, which is 9/2, plus 6 time 3, which is 18, and I have to subtract off what I get when I plug in -1 everywhere -1^3 is -1/3, minus 1^2 is going to , with a negative sign, so, minus a half, when I plug in this, I get -6. If you work all that out, I think you'll see 29 1/3. You can just check the arithmetic there, and so, this equals 29 1/3. And so therefore, the area of this interesting yellow region turns out to be 29 1/3 units squared. Okay, we'll take a look at another one of these examples together up next. See you there.

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