Question

# Evaluate the iterated integral \int_{-1}^{2}\int_{0}^{\frac{\pi}{2}}

Applications of integrals
Evaluate the iterated integral
$$\int_{-1}^{2}\int_{0}^{\frac{\pi}{2}}$$

2021-05-29
Step 1
Iterated integral is integral over several variables such that only one variable is integrated over at one time keeping the other variable as a constant. The iterated integral can be written as a product of two independent integrals if the limits of the integration are constants and the integrand can be separated.
An example of integrand which can be separated is xy. An example of integrand which cannot be separated is $$\sin(xy)$$.
Step 2
Given iterated integral to be computed is $$\int_{-1}^{2}\int_{0}^{\frac{\pi}{2}} y\sin x dxdy$$. The integrand can be separated and the limits are constants so the integral can be written as product of two integrals: one integrating over x and other integraing over y.
Use this information to compute the integral.
$$\int_{-1}^{2}\int_{0}^{\frac{\pi}{2}} y\sin x dxdy=(\int_{-1}^{2}y dy)(\int_{0}^{\frac{\pi}{2}}\sin x dx)$$
$$=(\frac{y^{2}}{2})_{-1}^{2}(-\cos x)_{0}^{\frac{\pi}{2}}$$
$$=(\frac{2^{2}-(-1)^{2}}{2})(-\cos \frac{\pi}{2}+\cos 0)$$
$$=\frac{3}{2}*1$$
$$=\frac{3}{2}$$
Hence, the given iterated integral is equal to $$\frac{3}{2}$$.