Solve ∫ (from 0 to 4) of (4-y) wrt y

$$\int _ { 0 } ^ { 4 } ( 4 - y ) d y$$

Evaluate

$$8$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4-y\mathrm{d}y$$

Integrate the sum term by term.

$$\int 4\mathrm{d}y+\int -y\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 4\mathrm{d}y-\int y\mathrm{d}y$$

Find the integral of 4 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$4y-\int y\mathrm{d}y$$

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 -1 times \frac{y^{2}}{2}.

$$4y-\frac{y^{2}}{2}$$

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.

$$4\times 4-\frac{4^{2}}{2}-\left(4\times 0-\frac{0^{2}}{2}\right)$$

Simplify.

$$8$$