Solve ∫ (from 0 to 2 pi) of pi(x^2)^2 wrt x

$$\int _ { 0 } ^ { 2 \pi } \pi ( x ^ { 2 } ) ^ { 2 } d x$$

Evaluate

$$\frac{32\pi ^{6}}{5}\approx 6152.890838882$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \pi \left(x^{2}\right)^{2}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\pi \int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}.

$$\pi \times \frac{x^{5}}{5}$$

Simplify.

$$\frac{\pi x^{5}}{5}$$

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.

$$\frac{1}{5}\pi \times \left(2\pi \right)^{5}-\frac{1}{5}\pi \times 0^{5}$$

Simplify.

$$\frac{32\pi ^{6}}{5}$$