Triple integrals over general regions practice problems
Triple integrals over regions bounded by planes and simple surfaces, one variable at a time. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Evaluate \( \displaystyle \iiint_E x z\, dV \), where \( \displaystyle E = \{(x, y, z) : -1 \le x \le 1,\ 0 \le y \le 1,\ 0 \le z \le 1 - x^{2}\} \).
Evaluate \( \displaystyle \iiint_E x + y + z\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 4,\ 0 \le y \le \sqrt{x},\ 0 \le z \le y\} \).
Evaluate \( \displaystyle \iiint_E x + 1\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 2,\ 0 \le y \le x,\ 0 \le z \le x + y\} \).
Evaluate \( \displaystyle \iiint_E y\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 1,\ 0 \le y \le \sqrt{x},\ 0 \le z \le y\} \).
Evaluate \( \displaystyle \iiint_E z\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 2,\ 0 \le y \le x,\ 0 \le z \le x + y\} \).
Evaluate \( \displaystyle \iiint_E z\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 1,\ 0 \le y \le x,\ 0 \le z \le x + y\} \).
Evaluate \( \displaystyle \iiint_E 1\, dV \), where \( \displaystyle E = \{(x, y, z) : -1 \le x \le 1,\ 0 \le y \le 1,\ 0 \le z \le 1 - x^{2}\} \).
Evaluate \( \displaystyle \iiint_E x + 2 y z\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 2,\ 0 \le y \le x,\ 0 \le z \le x + y\} \).
Evaluate \( \displaystyle \iiint_E z\, dV \), where \( \displaystyle E = \{(x, y, z) : -1 \le x \le 1,\ 0 \le y \le 3,\ 0 \le z \le 1 - x^{2}\} \).
Evaluate \( \displaystyle \iiint_E 1\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 3,\ 0 \le y \le 3 - x,\ 0 \le z \le - x - y + 3\} \).