Solve ∫ (from 0 to 11) of 625left(11-yright)10 wrt y

$$\int_{ 0 }^{ 11 } 625 \left( 11-y \right) 10 d y$$

Evaluate

$$378125$$

Solution Steps

Multiply 625 and 10 to get 6250.

$$\int _{0}^{11}6250\left(11-y\right)\mathrm{d}y$$

Use the distributive property to multiply 6250 by 11-y.

$$\int _{0}^{11}68750-6250y\mathrm{d}y$$

Evaluate the indefinite integral first.

$$\int 68750-6250y\mathrm{d}y$$

Integrate the sum term by term.

$$\int 68750\mathrm{d}y+\int -6250y\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 68750\mathrm{d}y-6250\int y\mathrm{d}y$$

Find the integral of 68750 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$68750y-6250\int y\mathrm{d}y$$

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 -6250 times \frac{y^{2}}{2}.

$$68750y-3125y^{2}$$

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.

$$68750\times 11-3125\times 11^{2}-\left(68750\times 0-3125\times 0^{2}\right)$$

Simplify.

$$378125$$