The integral π∫(1−y^2)^2 dy (low lim=-1 upp lim=1) gives the volume of a particular solid. Sketch the solid and find the area of that region in the xy-plane. Also find the volume of the solid.

Answer 1

So this is the curve x(y) = 1-y² where the area is the integral (from -1 to 1) of xdy and the volume is the curve rotated around the y axis: The integral (from -1 to 1) of πx²dy

Area integral: (1-y²) from -1 to 1 is basic

Volume integral π(1-y²)² from -1 to 1 is not bad if you multiply it out: pi(1-2y²+y⁴) dy from -1 to 1. Again, just the integral of a polynomial.