Solve ∫ (from 0 to 9) of (11-y)(20/9y) wrt y

$$\int _ { 0 } ^ { 9 } ( 11 - y ) ( \frac { 20 } { 9 } y ) d y$$

Evaluate

$$450$$

Solution Steps

Use the distributive property to multiply 11-y by \frac{20}{9}.

$$\int _{0}^{9}\left(11\times \frac{20}{9}-y\times \frac{20}{9}\right)y\mathrm{d}y$$

Express 11\times \frac{20}{9} as a single fraction.

$$\int _{0}^{9}\left(\frac{11\times 20}{9}-y\times \frac{20}{9}\right)y\mathrm{d}y$$

Multiply 11 and 20 to get 220.

$$\int _{0}^{9}\left(\frac{220}{9}-y\times \frac{20}{9}\right)y\mathrm{d}y$$

Multiply -1 and \frac{20}{9} to get -\frac{20}{9}.

$$\int _{0}^{9}\left(\frac{220}{9}-\frac{20}{9}y\right)y\mathrm{d}y$$

Use the distributive property to multiply \frac{220}{9}-\frac{20}{9}y by y.

$$\int _{0}^{9}\frac{220}{9}y-\frac{20}{9}yy\mathrm{d}y$$

Multiply y and y to get y^{2}.

$$\int _{0}^{9}\frac{220}{9}y-\frac{20}{9}y^{2}\mathrm{d}y$$

Evaluate the indefinite integral first.

$$\int \frac{220y-20y^{2}}{9}\mathrm{d}y$$

Integrate the sum term by term.

$$\int \frac{220y}{9}\mathrm{d}y+\int -\frac{20y^{2}}{9}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\frac{220\int y\mathrm{d}y-20\int y^{2}\mathrm{d}y}{9}$$

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 \frac{220}{9} times \frac{y^{2}}{2}.

$$\frac{110y^{2}-20\int y^{2}\mathrm{d}y}{9}$$

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 -\frac{20}{9} times \frac{y^{3}}{3}.

$$\frac{110y^{2}}{9}-\frac{20y^{3}}{27}$$

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.

$$\frac{110}{9}\times 9^{2}-\frac{20}{27}\times 9^{3}-\left(\frac{110}{9}\times 0^{2}-\frac{20}{27}\times 0^{3}\right)$$

Simplify.

$$450$$