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

Question
Integrals
Evaluate the following iterated integrals.
$$\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx$$

2021-01-09
$$\text{We have to evaluate the integral }$$
$$\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx$$
$$\int_1^3 x(\int_0^{\pi/2}\sin y\ dy)dx$$
$$\int_1^3 x[-\cos y]_0^{\pi/2}dx$$
$$-\int_1^3 x(0-1)dx$$
$$\int_1^3 xdx$$
$$[\frac{x^2}{2}]_1^3$$
$$=\frac{9}{2}-\frac{1}{2}$$
$$=\frac{8}{2}$$
$$=4$$
$$\text{therefore}$$
$$\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx=4$$

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