Compute the volume in the first octant

Question

Use a double integral to compute the volume in the first octant above the region in the xy plane bounded by y=1-x² , y=0, and x=0; and between the planes z=2-y and z=1

Answer 1

Use the integral over dzdydx where dz=2-y-1 and y ranges from 0 to 1-x² and x ranges from 0 to1. Doing this gives (2-y-1)dydx gives (1-y)dydx. The integral becomes (y-y²/2)₀^{1-x^2}dx. Substitute in the limits and get 1/2 -x⁴/2 which integrates to (x/2 - x⁵/10)¹₀ which gives the volume as 2/5.