Solve ∫ (from 0 to 1000) of sqrt{x wrt x

$$\int_{ 0 }^{ 1000 } \sqrt{ x } d x$$

Evaluate

$$\frac{20000\sqrt{10}}{3}\approx 21081.851067789$$

Solution Steps

Evaluate the indefinite integral first.

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

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.

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

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.

$$\frac{2}{3}\times 1000^{\frac{3}{2}}-\frac{2}{3}\times 0^{\frac{3}{2}}$$

Simplify.

$$\frac{20000\sqrt{10}}{3}$$