Solve ∫ (from - 1 to 2) of (4+2y-2y^2) wrt y

$$\int _ { - 1 } ^ { 2 } ( 4 + 2 y - 2 y ^ { 2 } ) d y$$

Evaluate

$$9$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4+2y-2y^{2}\mathrm{d}y$$

Integrate the sum term by term.

$$\int 4\mathrm{d}y+\int 2y\mathrm{d}y+\int -2y^{2}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 4\mathrm{d}y+2\int y\mathrm{d}y-2\int y^{2}\mathrm{d}y$$

Find the integral of 4 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$4y+2\int y\mathrm{d}y-2\int y^{2}\mathrm{d}y$$

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

$$4y+y^{2}-2\int y^{2}\mathrm{d}y$$

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 -2 times \frac{y^{3}}{3}.

$$4y+y^{2}-\frac{2y^{3}}{3}$$

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.

$$4\times 2+2^{2}-\frac{2}{3}\times 2^{3}-\left(4\left(-1\right)+\left(-1\right)^{2}-\frac{2}{3}\left(-1\right)^{3}\right)$$

Simplify.

$$9$$