Solve ∫ (from 0 to 11) of 62.5left(11-yright)10 wrt y

$$\int_{ 0 }^{ 11 } 62.5 \left( 11-y \right) 10 d y$$

Evaluate

$$37812.5$$

Solution Steps

Multiply 62.5 and 10 to get 625.

$$\int _{0}^{11}625\left(11-y\right)\mathrm{d}y$$

Use the distributive property to multiply 625 by 11-y.

$$\int _{0}^{11}6875-625y\mathrm{d}y$$

Evaluate the indefinite integral first.

$$\int 6875-625y\mathrm{d}y$$

Integrate the sum term by term.

$$\int 6875\mathrm{d}y+\int -625y\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 6875\mathrm{d}y-625\int y\mathrm{d}y$$

Find the integral of 6875 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$6875y-625\int y\mathrm{d}y$$

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 -625 times \frac{y^{2}}{2}.

$$6875y-\frac{625y^{2}}{2}$$

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.

$$6875\times 11-\frac{625}{2}\times 11^{2}-\left(6875\times 0-\frac{625}{2}\times 0^{2}\right)$$

Simplify.

$$\frac{75625}{2}$$