Another problem involving computing area with integrals

[haydn]

Homework Statement

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region.

x+y^2=6

x+y=0

Homework Equations

none, just using integrals...

The Attempt at a Solution

I sketch the two curves and find they intersect at x=-3 and x=2. Looking at it I want to integrate with respect to x.

y=\sqrt{-x+6} appears to be the top function, while y=-x is the lower function.

I set up an integral from -3 to 2 that consists of: \sqrt{-x+6} - (-x) dx.

I evaluate using substitution and keep getting 10.16, which the homework website I'm using says is wrong. Not sure what I'm doing incorrectly here.

 

[Mark44]

It's a lot easier if you integrate w.r.t y (i.e., use horizontal area elements rather than vertical elements). If you do it this way there are no radicals to deal with. The area I get is 49/3.

You can find the area using vertical area elements, but the y values at the bottoms of your area elements change. For one interval, the function at the bottom is y = -x; for the other interval, the function at the bottom is y = -sqrt(-x + 6). This means that your formula for the area of the typical element has to change, and that means you need to integrals.

 

[Mark44]

BTW, this problem is very similar to the one posted yesterday by vipertongn.

 

[haydn]

Hey thanks a lot. I get it now. I got the correct answer.