Solve ∫ (from 0 to 10) of 0.003p^3dp

$$\int _ { 0 } ^ { 10 } 0.003 p ^ { 3 } d p$$

Evaluate

$$7.5$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{3p^{3}}{1000}\mathrm{d}p$$

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

$$\frac{3\int p^{3}\mathrm{d}p}{1000}$$

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

$$\frac{3p^{4}}{4000}$$

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.

$$\frac{3}{4000}\times 10^{4}-\frac{3}{4000}\times 0^{4}$$

Simplify.

$$\frac{15}{2}$$