Solve ∫ (from 0 to 2 pi) of 4x^2 wrt x

$$\int _ { 0 } ^ { 2 \pi } 4 x ^ { 2 } d x$$

Evaluate

$$\frac{32\pi ^{3}}{3}\approx 330.733617923$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x^{2}\mathrm{d}x$$

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

$$4\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$\frac{4x^{3}}{3}$$

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.

$$\frac{4}{3}\times \left(2\pi \right)^{3}-\frac{4}{3}\times 0^{3}$$

Simplify.

$$\frac{32\pi ^{3}}{3}$$