Triple Integrals Practice

Desmos

Find Volume of the Tetrahedron Enclosed by $x=0,y-0,z=0$ $2x+y+Z=4$

Step 1: sketch both the solid and the integration region

Desmos

setting up the integral:

$${\int }_0^2{\int }_0^{4-2x}{\int }_0^{4-2x-y}dzdydx$$

Find the Volume of the Solid Enclosed by the Cylinder $y=x^2$ and the Planes $z=0$ and $y+z=1$

Desmos

$$\int \int \int dzdsomn$$ $${\int }_{-1}^1{\int }_{x^2}^1{\int }_0^{1-y}dzdydx$$

evaluate the triple integral $\int {\int }_B\int (x^2+y^2+z^2{)}^{2}dV$ with a sphere at the center origin, with a radius of 3

$$x^2+y^2+z^2={\rho }^2$$ $${\int }_0^{2\pi }{\int }_0^{\pi }{\int }_0^3({\rho }^2{)}^2{\rho }^2\sin$$