Solve ∫ (from 0 to 2) of y+2-(y^2) wrt y

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

Evaluate

$$\frac{10}{3}\approx 3.333333333$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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

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

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}.

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

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.

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

Simplify.

$$\frac{10}{3}$$