Solve ∫ (from 2 to 4) of pix^3 wrt x

$$\int _ { 2 } ^ { 4 } \pi x ^ { 3 } d x$$

Evaluate

$$60\pi \approx 188.495559215$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \pi x^{3}\mathrm{d}x$$

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

$$\pi \int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

$$\pi \times \frac{x^{4}}{4}$$

Simplify.

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

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.

$$\frac{1}{4}\pi \times 4^{4}-\frac{1}{4}\pi \times 2^{4}$$

Simplify.

$$60\pi$$