Solve ∫ (from 0 to frac { 1340) of {90}}(1340-27x) wrt x

$$\int _ { 0 } ^ { \frac { 1340 } { 90 } } ( 1340 - 27 x ) d x$$

Evaluate

$$\frac{152626}{9}\approx 16958.444444444$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1340-27x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1340\mathrm{d}x+\int -27x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1340\mathrm{d}x-27\int x\mathrm{d}x$$

Find the integral of 1340 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$1340x-27\int x\mathrm{d}x$$

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 -27 times \frac{x^{2}}{2}.

$$1340x-\frac{27x^{2}}{2}$$

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.

$$1340\times \frac{134}{9}-\frac{27}{2}\times \left(\frac{134}{9}\right)^{2}-\left(1340\times 0-\frac{27}{2}\times 0^{2}\right)$$

Simplify.

$$\frac{152626}{9}$$