Trip Integral

#416 / Math Hard

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

Expand 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 $$x$$ :
$$$=\int_1^3 \frac{ 2x^4z^2}{4}\bigg\rvert_{x=-1}^{x=1}\: dz=\int_1^3 \frac{(1)^4z^2}{2}-\frac{(-1)^4z^2}{2}\: dz$$$
$$$=\int_1^3 \frac{z^2}{2}-\frac{z^2}{2}\: dz=\int_1^3 0\: dz=0$$$
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