Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } ( x + 3 \sqrt { y } ) d y d x$ .
A)10
B)20
C)8
D)4
E)12
F)6
G)7
H)14

Question

Quizplus · Accepted Answer

First, integrate with respect to $ y $: $\int _ { 0 } ^ { x ^ { 2 } } ( x + 3 \sqrt { y } ) d y = xy + 2y^{3/2} \bigg|_0^{x^2} = x(x^2) + 2(x^2)^{3/2} = x^3 + 2x^3 = 3x^3$. Then, integrate with respect to $ x $: $\int _ { 0 } ^ { 2 } 3x^3 dx = \frac{3}{4}x^4 \bigg|_0^2 = \frac{3}{4}(16) = 12$.

Evaluate the Iterated Integral ∫02∫0x2(x+3y)dydx\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } ( x + 3 \sqrt { y } ) d y d x∫02​∫0x2​(x+3y​)dydx

Evaluate the Iterated Integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } ( x + 3 \sqrt { y } ) d y d x$