Answer to Question #139337 in Calculus for Promise Omiponle

Question

Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 6.

Answer 1

$V=\iint_D zdydx=\iint_D (6-x-y)dydx$

$D:0\leq x\leq 6, 0\leq y\leq 6-x$

$V=\int_0^6 \int_0^{6-x}(6-x-y)dydx =\int _0^6 (6y-xy-\frac{y^2}{2})|_0^{6-x}dx=$

$=\int_0 ^6 (6\cdot(6-x)-x\cdot(6-x)-\frac{(6-x)^2}{2})dx=\int_0^6((6-x)^2-\frac{(6-x)^2}{2})dx=$

$=\frac{1}{2} \int_0^6 (6-x)^2dx=\frac{1}{2} \int_0^6 (x-6)^2d(x-6)=(\frac{1}{2}\cdot \frac{(x-6)^3}{3} )|_0^6 =$

$=\frac{(6-6)^3}{6}-\frac{(0-6)^3}{6}=36$