Solve ∫ (from 0 to 4) of (4sqrt{x}) wrt x

$$\int _ { 0 } ^ { 4 } ( 4 \sqrt { x } ) d x$$

Evaluate

$$\frac{64}{3}\approx 21.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4\sqrt{x}\mathrm{d}x$$

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

$$4\int \sqrt{x}\mathrm{d}x$$

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

$$\frac{8x^{\frac{3}{2}}}{3}$$

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.

$$\frac{8}{3}\times 4^{\frac{3}{2}}-\frac{8}{3}\times 0^{\frac{3}{2}}$$

Simplify.

$$\frac{64}{3}$$