Area: horizontal areas between curves
Horizontal area between curves
- [Instructor] So I have two curves graphed here. And we're used to seeing things where y is a function of x, but here we have x as a function of y. In fact, we can write this top expression as being a function of y. And this second one, just to make it different, we could view this as g of y. Once again, it is a function of y. And what we're concerning ourselves with in this video is how do we find this area in this light blue color between these two curves? And I encourage you to pause the video and try to work through it. All right, so a huge hint here is we're going to want to integrate with respect to y, a definite integral where our bounds are in terms of y. So for example, this is, this lower point of intersection right over here, this would be our lower bound in terms of y. Let's call that y one. And then, this up here, this would be our upper y bound. So if we think about where do these two curves intersect, and we look at the y-coordinates of those intersections, well, that gives us two nice bounds for our integral. So we're gonna take our integral from y one to y two, from y one to y two, y two. We're going to integrate with respect to y, dy. And so what are we going to sum up? Well, when we integrate, we can think about taking the sum of infinitely thin rectangles. And in this case, it would infinitely flat rectangles since we're thinking about dy. So dy would be the height of each of these rectangles. And what would be, in this case, the width or the length of this rectangle right over here? Well, over this interval, from y one to y two, our blue function, f of y, takes on larger x-values than g of y. So this length right over here, this would be f of y, f of y, this x-value minus this x-value, minus g of y. So this is going to be f of y minus g of y, g of y. Well, we know what f of y and g of y are. Really, the trickiest part is, is figuring out these points of intersection. So let's think about where these two curves intersect. They are both equal to x, so we can set these two y expressions equal to each other. So we know that negative, let me do it in that other color, so we know that negative y squared plus three y plus 11 is going to be equal to this, is going to be equal to y squared plus y minus one. So let's just subtract all of this from both sides so that on the right side we have a zero and on the left side we just have a quadratic. So let's subtract y squared. Let's subtract y. And then subtract negative one, which is just adding one. And over here, we're going to do the same thing, minus y plus one. And what we are left with is going to be a, hopefully, a straightforward quadratic. So let's see, this is going to be negative two y squared plus two y. Am I doing that right? Yep, plus two y, plus 12 is equal to zero. And then this over here I can factor out a negative two, and I get negative two times y squared minus y minus six is equal to zero. This we can factor from inspection. What two numbers, when we add, equal negative one? When we take their product, we get negative six? Well, that would be negative three and two. So this is going to be negative two times y minus three times y plus two. That's just straightforward factoring a polynomial, a quadratic. Did I do that right? Yep, that looks right, is equal to zero. So what are the points of intersection? The points of intersection are going to be y is equal to three and y is equal to negative two. So this right over here is y is equal to negative two, and then the upper bound is y is equal to three. So now we just have to evaluate this from negative two, all the way until three. So let's do that. I'm gonna clear this out, so I get a little bit of real estate. So this is equal to the integral from negative two to three of negative y squared, plus three y, plus 11, minus all of this stuff. So if we just distribute a negative sign here, it's minus y squared, minus y, plus one, and then we have a dy, dy. This is equal to the definite integral from negative two to positive three of, let's see, negative y squared minus y squared, negative two y squared, and then three y minus y is going to be plus two y. And then 11 plus one, plus 12, we saw this just now, when we were trying to solve for y, dy. And so what is that going to be equal to? Well, we just take the antiderivative here. This is going to be, let's see, negative two. Let's increment the exponent, y to the third. Divide by that exponent, reverse power rule, plus two y squared divided by two, which is just y squared, just the reverse power rule, and then plus 12 y. And we're going to evaluate that at three and at negative two. So if we evaluate that at three, we are going to get, let's see, negative two times 27 over three, plus nine, plus 36. And then we are going to want to subtract, minus all of this evaluated at negative two. So it's going to be negative two times negative eight over three, plus four, minus 24. So we just have a little bit of mathematics ahead of us. So let's see, this is going to be 27 divided by three is nine. So this is negative 18. Negative 18 plus nine is going to be negative nine, plus 36, all of that is going to be equal to, so the stuff in blue is equal to 27. Right, did I do that right? Get negative 18 plus nine, yep, 27. And then all the stuff in red over here, we have, this is going to be negative times a negative, so it's 16 over three, plus four, minus 24. So that is going to be 16 over three, and then minus, minus 20, minus 20. But then we have this negative out here, so it's, if we distribute that, we'll get plus 20, minus, we could say instead of 16 over three, we could rewrite that as 5 1/3, minus 5 1/3. And so what is that going to get us? Let me scroll down a little bit, or let me go to the right, so I have a little bit more real estate here. So then that is going to be equal to, get a minor drum roll here, 47 minus five minus 1/3, which is equal to, let's see, 47 minus five, that's 42, minus 1/3 is, now we get a drum roll, 41 2/3. And we are done.