Solve ∫ (from -1 to 1) of -y^2-1 wrt y

$$\int_{ -1 }^{ 1 } - { y }^{ 2 } -1 d y$$

Evaluate

$$-\frac{8}{3}\approx -2.666666667$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

$$-\frac{y^{3}}{3}+\int -1\mathrm{d}y$$

Find the integral of -1 using the table of common integrals rule \int a\mathrm{d}y=ay.

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

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.

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

Simplify.

$$-\frac{8}{3}$$