## Trip Integral

Evaluate the shown triple integral where B = {(x,y,z) | -1 ≤ x ≤ 1, 0 ≤ y ≤ 2, 1 ≤ z ≤ 3}.

Hint
The integration order is not specified, but we can use the iterated integral in any order without changing the difficulty.
Hint 2
$$\int x^n \:dx=\frac{x^{n+1}}{n+1}$$$The integration order is not specified, but we can use the iterated integral in any order without changing the difficulty. Let’s arbitrarily integrate $$y$$ first, then $$x$$ , and then finally $$z$$ . Recall the power rule: $$\int x^n \:dx=\frac{x^{n+1}}{n+1}$$$
Integrate with respect to $$y$$ :
$$\int_1^3 \int_{-1}^1 \int_0^2 x^3yz^2\:dy\, dx\, dz$$$$$=\int_1^3 \int_{-1}^1 \frac{x^3y^2z^2}{2}\bigg\rvert_{y=0}^{y=2}\: dx\, dz=\int_1^3 \int_{-1}^1\frac{x^32^2z^2}{2}-0\: dx\, dz=\int_1^3 \int_{-1}^12x^3z^2\: dx\, dz$$$
Integrate with respect to $$x$$ :
$$=\int_1^3 \frac{ 2x^4z^2}{4}\bigg\rvert_{x=-1}^{x=1}\: dz=\int_1^3 \frac{(1)^4z^2}{2}-\frac{(-1)^4z^2}{2}\: dz$$$$$=\int_1^3 \frac{z^2}{2}-\frac{z^2}{2}\: dz=\int_1^3 0\: dz=0$$$
0