Question

Find the volume of the region bounded by the paraboloid z=x2+y2 and the plane x+y=1 in the first octant.

Answer 1

Answer:

Step-by-step explanation:

Given

Paraboloid $z=x^2+y^2$ and Plane $x+y=1$ are bounded to get the area

In first octant

value of x varies from $0\leq x\leq 1$

value of y varies from $0\leq y\leq 1$

Volume $V=\int_{0}^{1}\int_{0}^{1-x}\left ( x^2+y^2\right )dydx$

$V=\int_{0}^{1}\left [ x^2y+\frac{y^3}{3}\right ]_{0}^{1-x}$

$V=\int_{0}^{1}\left [ x^2(1-x)+\frac{(1-x)^3}{3}\right ]dx$

$V=\int_{0}^{1}\left ( \frac{-4x^3}{3}+2x^2-x+\frac{1}{3}\right )dx$

$V=\frac{1}{6}$