Solve ∫ (from 10 to 20) of 80-4x wrt x

$$\int_{ 10 }^{ 20 } 80-4x d x$$

Evaluate

$$200$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 80-4x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 80\mathrm{d}x+\int -4x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 80\mathrm{d}x-4\int x\mathrm{d}x$$

Find the integral of 80 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$80x-4\int x\mathrm{d}x$$

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 -4 times \frac{x^{2}}{2}.

$$80x-2x^{2}$$

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.

$$80\times 20-2\times 20^{2}-\left(80\times 10-2\times 10^{2}\right)$$

Simplify.

$$200$$