Solve ∫ (from - 1 to 1) of (1-y)y wrt y

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

Evaluate

$$-\frac{2}{3}\approx -0.666666667$$

Solution Steps

Use the distributive property to multiply 1-y by y.

$$\int _{-1}^{1}y-y^{2}\mathrm{d}y$$

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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

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

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.

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

Simplify.

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