Solve ∫ (from 18 to 20) of 0.1(20-x) wrt x

$$\int _ { 18 } ^ { 20 } 0.1 ( 20 - x ) d x$$

Evaluate

$$0.2$$

Solution Steps

Use the distributive property to multiply 0.1 by 20-x.

$$\int _{18}^{20}2-0.1x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 2-\frac{x}{10}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2\mathrm{d}x+\int -\frac{x}{10}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 2\mathrm{d}x-\frac{\int x\mathrm{d}x}{10}$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2x-\frac{\int x\mathrm{d}x}{10}$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply -0.1 times \frac{x^{2}}{2}.

$$2x-\frac{x^{2}}{20}$$

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.

$$2\times 20-\frac{20^{2}}{20}-\left(2\times 18-\frac{18^{2}}{20}\right)$$

Simplify.

$$\frac{1}{5}$$