Evaluate the improper integral.
-$$\int _ { 1 } ^ { 5 } \int _ { 2 } ^ { 218 } \frac { d y d x } { \sqrt { ( x - 1 ) } ( y - 2 ) ^ { 2 / 3 } }$$
A) 60
B) 36
C) 48
D) 72

Question

Quizplus · Accepted Answer

We first integrate with respect to $y$ using the substitution $u=(y-2)^{1/3}$, which gives us 
$$\int _ { 1 } ^ { 5 } 3u\,du\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x - 1 } } \, dx.$$
Evaluating the $x$ integral with the substitution $t=x-1$ gives us $\int_{0}^{9}\frac{1}{\sqrt{t}}\,dt=6$. Thus, the entire integral is equal to $3\cdot 6\cdot\int_{1}^{5}u\,du = 72$. Therefore, the answer is D.

Evaluate the Improper Integral ∫15∫2218dydx(x−1)(y−2)2/3\int _ { 1 } ^ { 5 } \int _ { 2 } ^ { 218 } \frac { d y d x } { \sqrt { ( x - 1 ) } ( y - 2 ) ^ { 2 / 3 } }∫15​∫2218​(x−1)​(y−2)2/3dydx​

Evaluate the Improper Integral $\int _ { 1 } ^ { 5 } \int _ { 2 } ^ { 218 } \frac { d y d x } { \sqrt { ( x - 1 ) } ( y - 2 ) ^ { 2 / 3 } }$