Calculate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { x } ( x - 1 ) d y d x$
A) - $\frac { 1 } { 12 }$
B) - $\frac { 1 } { 2 }$
C) - $\frac { 3 } { 4 }$
D) $\frac { 1 } { 2 }$
E) none of these

Question

Quizplus · Accepted Answer

The integral simplifies as follows: $\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { x } ( x - 1 ) d y d x = \int _ { 0 } ^ { 1 } ( x - 1 )( x - x ^ { 2 } ) d x = \int _ { 0 } ^ { 1 } ( x ^ { 2 } - x ^ { 3 } - x + x ^ { 2 } ) d x = \int _ { 0 } ^ { 1 } ( 2 x ^ { 2 } - x ^ { 3 } - x ) d x$. Integrating term by term gives $\frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 } - \frac { 1 } { 2 } x ^ { 2 }$ evaluated from 0 to 1, which equals $\frac { 2 } { 3 } - \frac { 1 } { 4 } - \frac { 1 } { 2 } = \frac { 8 } { 12 } - \frac { 3 } { 12 } - \frac { 6 } { 12 } = - \frac { 1 } { 12 }$.

Calculate the Iterated Integral ∫01∫x2x(x−1)dydx\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { x } ( x - 1 ) d y d x∫01​∫x2x​(x−1)dydx

Calculate the Iterated Integral $\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { x } ( x - 1 ) d y d x$