Solve ∫ (from 0 to 1) of (2-y-sqrt{y}) wrt y

$$\int _ { 0 } ^ { 1 } ( 2 - y - \sqrt { y } ) d y$$

Evaluate

$$\frac{5}{6}\approx 0.833333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2-y-\sqrt{y}\mathrm{d}y$$

Integrate the sum term by term.

$$\int 2\mathrm{d}y+\int -y\mathrm{d}y+\int -\sqrt{y}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 2\mathrm{d}y-\int y\mathrm{d}y-\int \sqrt{y}\mathrm{d}y$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$2y-\int y\mathrm{d}y-\int \sqrt{y}\mathrm{d}y$$

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

$$2y-\frac{y^{2}}{2}-\int \sqrt{y}\mathrm{d}y$$

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

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

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.

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

Simplify.

$$\frac{5}{6}$$