Solve ∫ (from 0 to 1) of 135(2-h)dh

$$\int _ { 0 } ^ { 1 } 135 ( 2 - h ) d h$$

Evaluate

$$\frac{405}{2}=202.5$$

Solution Steps

Use the distributive property to multiply 135 by 2-h.

$$\int _{0}^{1}270-135h\mathrm{d}h$$

Evaluate the indefinite integral first.

$$\int 270-135h\mathrm{d}h$$

Integrate the sum term by term.

$$\int 270\mathrm{d}h+\int -135h\mathrm{d}h$$

Factor out the constant in each of the terms.

$$\int 270\mathrm{d}h-135\int h\mathrm{d}h$$

Find the integral of 270 using the table of common integrals rule \int a\mathrm{d}h=ah.

$$270h-135\int h\mathrm{d}h$$

Since \int h^{k}\mathrm{d}h=\frac{h^{k+1}}{k+1} for k\neq -1, replace \int h\mathrm{d}h with \frac{h^{2}}{2}. Multiply -135 times \frac{h^{2}}{2}.

$$270h-\frac{135h^{2}}{2}$$

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.

$$270\times 1-\frac{135}{2}\times 1^{2}-\left(270\times 0-\frac{135}{2}\times 0^{2}\right)$$

Simplify.

$$\frac{405}{2}$$