# Area between two curves integral

## Homework Statement

Consider the region enclosed by the curves x=2-y^2 and y=-x

Write a single integral that can be used to evaluate the area of the region. Find this area. Your answer should be a fraction reduced to its lowest terms.

NA

## The Attempt at a Solution

First, I graphed them and found the intersection points. The graphs intersect at the points (-2,2) and (1,-1). To write the integral, I decided to integrate with respect to y, I have:

Integral from y=2 to y=-1 of (-y)-(2-y^2)dy

Solving this, I got that the area should be 1.5, however I was told the answer is 4.5. Any ideas where I went wrong?

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Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

Consider the region enclosed by the curves x=2-y^2 and y=-x

Write a single integral that can be used to evaluate the area of the region. Find this area. Your answer should be a fraction reduced to its lowest terms.

NA

## The Attempt at a Solution

First, I graphed them and found the intersection points. The graphs intersect at the points (-2,2) and (1,-1). To write the integral, I decided to integrate with respect to y, I have:

Integral from y=2 to y=-1 of (-y)-(2-y^2)dy

Solving this, I got that the area should be 1.5, however I was told the answer is 4.5. Any ideas where I went wrong?
Remember: an integral is the limit of a sum of a large number of small quantities. When you evaluate a planar area you can either split it up into (a large number of narrow) vertical rectangles (long sides parallel to the y-axis) or horizontal rectangles (long sides parallel to the x-axis). Which method have you attempted to use? Did you do it correctly?

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DrClaude
Mentor
Integral from y=2 to y=-1 of (-y)-(2-y^2)dy

Solving this, I got that the area should be 1.5, however I was told the answer is 4.5. Any ideas where I went wrong?
That should work. You'll have to give the details of the integration.

Well you have set this up in a very unusual form (for example, why is your lower limit 2 and your upper limit -1?), but it is a correct setup and as it turns out the integral from y=2 to y=-1 of (-y)-(2-y^2) dy is indeed 4.5. So it looks like you're having trouble with evaluating the integral.

If you need more help, why not show how you tried to evaluate the integral.

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Thank you, I realized I made a stupid mistake. - I made a stupid mistake taking the integral.. Thank you guys.