If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Area between a curve and the 𝘺-axis

AP.CALC:
CHA‑5 (EU)
,
CHA‑5.A (LO)
,
CHA‑5.A.2 (EK)

## Video transcript

so right over here I have the graph of the function y is equal to 15 over X or at least I see the part of it for positive values of X now what I'm curious about in this video is I want to find the area not between this curve and the positive x axis I want to find the area between the curve and the y-axis bounded not by 2 X values but bounded by two Y values so with the bottom bound of the horizontal line y is equal to e and an upper bound with y is equal to e to the third power so pause this video and see if you can work through it so one way to think about it this is just like definite integrals we've done where we're looking between the curve and the x-axis but now it looks like things are swapped around we now care about the y-axis so let's just rewrite our function here and let's rewrite it in terms of X so Y is equal to 15 over X that means if we multiply both sides by X X Y is equal to 15 and if we divide both sides by Y we get X is equal to 15 over Y these right over here are all going to be equivalent now how do this right over here help you well think about the area think about estimating the area is a bunch of little rectangles here so that's one rectangle and then another rectangle right over there and then another rectangle right over there so what's the area of each of those rectangles so the width here that is going to be X but we can express X as a function of Y so that's the width right over there and we know that that's going to be 15 over Y and then what's the height going to be well it's going to be a very small change in Y the height is going to be dy so the area of one of those little rectangles right over there say the area of that one right over there you could view as it over here as 15 over y dy and then we want to sum all of these little rectangles from y is equal to e all the way to Y is equal to e to the third power so that's what our definite integral does we go from Y is equal to e the Y is equal to e to the third power so all we did we're used to seeing things like this where this would be 15 over X DX all we're doing here is this is 15 over Y D Y so let's evaluate this so we take the antiderivative of 15 over Y and then evaluated these two points so this is going to be equal to antiderivative of 1 over Y is going to be the natural log of the absolute value of y so it's 15 times the natural log of the absolute value of y and then we're going to evaluate that at our endpoints so we're going to evaluate it at e to the third and at E so let's first evaluate it e to the third so that's 15 times the natural log the absolute times the maps lateral log of the absolute value of e to the third power minus 15 times the natural log of the absolute value of e so what does this simplify to the natural log of e to the third power what power would have to raise e to to get to e to the third well that's just going to be three and then the natural log of e what power do after is e to to get e well that's just 1 so this is 15 times 3 minus 15 so that is all going to get us to 30 and we are done 45 minus 15
AP® is a registered trademark of the College Board, which has not reviewed this resource.