Solve ∫ (from 0 to 10) of pi^2z^4 wrt z

$$\int_{ 0 }^{ 10 } { \pi }^{ 2 } { z }^{ 4 } d z$$

Evaluate

$$20000\pi ^{2}\approx 197392.088021787$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \pi ^{2}z^{4}\mathrm{d}z$$

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

$$\pi ^{2}\int z^{4}\mathrm{d}z$$

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

$$\pi ^{2}\times \frac{z^{5}}{5}$$

Simplify.

$$\frac{\pi ^{2}z^{5}}{5}$$

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.

$$\frac{1}{5}\pi ^{2}\times 10^{5}-\frac{1}{5}\pi ^{2}\times 0^{5}$$

Simplify.

$$20000\pi ^{2}$$