Evaluate the iterated integral. ∫−11∫0πx2sin⁡ydydx

Question

Evaluate the iterated integral.  ∫−11∫0πx2sin⁡ydydx

Fachur · Accepted Answer

Step 1
For iterated integrals of the form

\int_{a}^{b} \int_{c}^{d} f (x) g (y) d y d x

. They can be simplified to

(\int_{a}^{b} f (x) d) (\int_{c}^{d} g (x) d x)

.
The given iterated integral is of this form. Note that this type of integral can be simplified one because the limits are not functions of x,y and second because the integrand function can be separated.
Step 2
The given integral is

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

. Simplify this integral into a product of two single integral and compute the integral.

\int_{- 1}^{1} \int_{0}^{π} x^{2} \sin (y) d y d x = (\int_{- 1}^{1} x^{2} d x) (\int_{0}^{π} \sin (y) d y)

= {(\frac{x^{3}}{3})}_{- 1}^{1} {(- \cos (y))}_{0}^{π}

= (\frac{2}{3}) (- \cos (π) + \cos (0))

= \frac{2}{3} (1 + 1)

= \frac{4}{3}

Evaluate the iterated integral. int_{-1}^{1}\int_{0}^{\pi}x^{2}

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