Solve ∫ (from 0 to 10) of -3sqrt{x wrt x

$$\int_{ 0 }^{ 10 } -3 \sqrt{ x } d x$$

Evaluate

$$-20\sqrt{10}\approx -63.245553203$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -3\sqrt{x}\mathrm{d}x$$

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

$$-3\int \sqrt{x}\mathrm{d}x$$

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. Multiply -3 times \frac{2x^{\frac{3}{2}}}{3}.

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

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.

$$-2\times 10^{\frac{3}{2}}+2\times 0^{\frac{3}{2}}$$

Simplify.

$$-20\sqrt{10}$$