Question

A solid right​ (noncircular) cylinder has its base R in the​ xy-plane and is bounded above by the paraboloid zequalsx2plusy2. The​ cylinder's volume is VequalsIntegral from 0 to 1 Integral from 0 to y (x squared plus y squared )dx dyplusIntegral from 1 to 2 Integral from 0 to 2 minus y (x squared plus y squared )dx dy Sketch the base of the region R and express the​ cylinder's volume as a single iterated integral with the order of integration reversed. Then evaluate the integral to find the volume.

Answer 1

a. V = 4/3

b. See attachment

Explanation:

a.

Given

Z = x² + y²

V = ∫ ∫ (x² + y²) dxdy {0,1}{0,y} + ∫ ∫ (x² + y²) dxdy {0,1}{x,2-x}

V = ∫ ∫ (x² + y²) dxdy {0,1}{x,2-x}

Integrate with respect to y

V = ∫ x²y+ y³/3 dx {0,1}{x,2-x}

V = ∫ x²(2-x) + (2-x)³/3 - x²(2) - (2)³/3 dx {0,1}

V = ∫ 2x² -7x³/3 + (2-x)³/3 dx {0,1}

V = 2x³/3 - 7x⁴/12 + (2-x)⁴/12 {0,1}

V = (⅔ - 7/4 + 2/12) - (0-0+16/12)

V = 4/3