Question

Sketch the region enclosed by the curves x=64−y^2 and x=y^2−64. Decide whether to integrate with respect to x or y. Then find the area of the region.
Area =

Answer 1

I assume you sketched the two curves.

You should have two horizontal parabolas that have the x-axis as their axes of symmetry, and y-intercepts of (0, ±8) and x-intercepts of (±64,0)

best way: Due to the nice symmetry, find the area of the region in the first quadrant and multiply that by 4, by using horizontal slices.
Area = 4 ∫ x dy from 0 to 8
= 4∫ (64-y^2) dy from 0 to 8
= 4 [64y - y^3/3] from 0 to 8
= 4(512 - 512/3 - (0-0) )
= 4(1024/3)
= 4096/3

check my arithmetic