Solve ∫ (from 0 to 1) of sqrt{x}-x^2

$$\int _ { 0 } ^ { 1 } \sqrt { x } - x ^ { 2 }$$

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{x}-x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \sqrt{x}\mathrm{d}x+\int -x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \sqrt{x}\mathrm{d}x-\int x^{2}\mathrm{d}x$$

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.

$$\frac{2x^{\frac{3}{2}}}{3}-\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -1 times \frac{x^{3}}{3}.

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

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.

$$\frac{2}{3}\times 1^{\frac{3}{2}}-\frac{1^{3}}{3}-\left(\frac{2}{3}\times 0^{\frac{3}{2}}-\frac{0^{3}}{3}\right)$$

Simplify.

$$\frac{1}{3}$$