Evaluate the iterated integral.
$$\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$$
A) 96
B) 116
C) 106
D) 126
E) 101

Question

Quizplus · Accepted Answer

First, integrate with respect to $x$: $\int_{y}^{3} 12xy \, dx = 12y \left[\frac{x^2}{2}\right]_{y}^{3} = 12y \left(\frac{9}{2} - \frac{y^2}{2}\right)$. Simplify to get $54y - 6y^3$. Now, integrate with respect to $y$: $\int_{1}^{3} (54y - 6y^3) \, dy = \left[27y^2 - \frac{3y^4}{2}\right]_{1}^{3}$. Evaluate to get $27(9) - \frac{3(81)}{2} - (27(1) - \frac{3(1)}{2}) = 243 - 121.5 - 25.5 = 96$.

Evaluate the Iterated Integral ∫13∫y312xydxdy\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y∫13​∫y3​12xydxdy

Evaluate the Iterated Integral $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$