Evaluate the following iterated integrals. int_1^3int_0^{pi/2}xsin y dy dx

Question

Evaluate the following iterated integrals.

\int_{1}^{3} \int_{0}^{π / 2} x \sin y d y d x

Answer 1

We have to evaluate the integral

\int_{1}^{3} \int_{0}^{π / 2} x \sin y d y d x

\int_{1}^{3} x (\int_{0}^{π / 2} \sin y d y) d x

\int_{1}^{3} x [- \cos y]_{0}^{π / 2} d x

- \int_{1}^{3} x (0 - 1) d x

\int_{1}^{3} x d x

[\frac{x^{2}}{2}]_{1}^{3}

= \frac{9}{2} - \frac{1}{2}

= \frac{8}{2}

= 4

therefore

\int_{1}^{3} \int_{0}^{π / 2} x \sin y d y d x = 4

Expert Answer