Solve ∫ (from 0 to pi) of int_0^1x^2(x^2+y^2)^3rdrdθ

$$\int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 1 } x ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) ^ { 3 } r d r d \theta$$

Evaluate

$$\frac{\pi x^{2}\left(x^{2}+y^{2}\right)^{3}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{0}^{1}x^{2}\left(x^{2}+y^{2}\right)^{3}r\mathrm{d}r\mathrm{d}\theta$$

Find the integral of \int _{0}^{1}x^{2}\left(x^{2}+y^{2}\right)^{3}r\mathrm{d}r using the table of common integrals rule \int a\mathrm{d}\theta =a\theta .

$$\int _{0}^{1}x^{2}\left(x^{2}+y^{2}\right)^{3}r\mathrm{d}r\theta$$

Simplify.

$$\frac{x^{2}\left(x^{2}+y^{2}\right)^{3}\theta }{2}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{2}x^{2}\left(x^{2}+y^{2}\right)^{3}\pi -\frac{1}{2}x^{2}\left(x^{2}+y^{2}\right)^{3}\times 0$$

Simplify.

$$\frac{x^{2}\left(x^{2}+y^{2}\right)^{3}\pi }{2}$$

Differentiate w.r.t. x

$$\pi x\left(4x^{2}+y^{2}\right)\left(x^{2}+y^{2}\right)^{2}$$