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

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

Evaluate

$$40$$

Solution Steps

Use the distributive property to multiply 11-y by 20.

$$\int _{9}^{11}220-20y\mathrm{d}y$$

Evaluate the indefinite integral first.

$$\int 220-20y\mathrm{d}y$$

Integrate the sum term by term.

$$\int 220\mathrm{d}y+\int -20y\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 220\mathrm{d}y-20\int y\mathrm{d}y$$

Find the integral of 220 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$220y-20\int y\mathrm{d}y$$

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 -20 times \frac{y^{2}}{2}.

$$220y-10y^{2}$$

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.

$$220\times 11-10\times 11^{2}-\left(220\times 9-10\times 9^{2}\right)$$

Simplify.

$$40$$