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

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

Evaluate

$$\frac{4}{3}\approx 1.333333333$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

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

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.

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

Simplify.

$$\frac{4}{3}$$