20 Important Questions on Triple Integration
Basic Cartesian Integrals
1. Evaluate:
$$\iiint_V (x+y+z)\, dV$$
where $0 \le x \le 1$, $0 \le y \le 2$, $0 \le z \le 3$.
2. Evaluate:
$$\iiint_V xyz \, dV$$
where $0 \le x \le 2$, $0 \le y \le 1$, $0 \le z \le 3$.
3. Evaluate:
$$\int_0^1 \int_0^{2x} \int_0^{x+y} dz\, dy\, dx$$
4. Change the order of integration:
$$\int_0^1 \int_0^{1-x} \int_0^{1-x-y} f(x,y,z)\, dz\, dy\, dx$$
Region Between Surfaces
5. Evaluate:
$$\iiint_V z \, dV$$
where $z = 0$ and $z = 4 - x^2 - y^2$.
6. Find the volume bounded by
$$z = x^2 + y^2 \quad \text{and} \quad z = 4$$
7. Evaluate:
$$\iiint_V (x^2+y^2)\, dV$$
inside cylinder $x^2 + y^2 = 9$, $0 \le z \le 5$.
8. Find the mass of the cube
$0 \le x,y,z \le 1$
with density $\rho = x+y+z$.
Spherical Coordinates
9. Find the volume of sphere
$$x^2+y^2+z^2 \le a^2$$
10. Evaluate:
$$\iiint_V (x^2+y^2+z^2)\, dV$$
over sphere of radius $R$.
11. Find the volume of upper hemisphere
$$x^2+y^2+z^2=16$$
Cylindrical Coordinates
12. Evaluate:
$$\iiint_V r \, dV$$
where $x^2+y^2=4$, $0 \le z \le 3$.
13. Find volume enclosed by
$$z=9-x^2-y^2$$ and $z=0$.
14. Evaluate:
$$\iiint_V z\, dV$$
inside cylinder $x^2+y^2=1$, $0 \le z \le 2$.
Change of Variables
15. Evaluate using $u=x+y$, $v=x-y$:
$$\iiint_V (x-y)\, dV$$
16. Evaluate:
$$\iiint_V e^{-(x^2+y^2+z^2)} dV$$
over entire space.
Applications
17. Find centroid of solid
$0 \le x \le a$, $0 \le y \le b$, $0 \le z \le c$.
18. Find moment of inertia about $z$-axis
for cylinder $x^2+y^2 \le R^2$, $0 \le z \le h$.
19. Evaluate over tetrahedron
$x=0$, $y=0$, $z=0$, $x+y+z=1$:
$$\iiint_V xyz\, dV$$
20. Find volume common to sphere
$$x^2+y^2+z^2=9$$
and cylinder $x^2+y^2=4$.
