Solve ∫ (from 0 to 3) of 2pit(100e^-0.55t)

$$\int _ { 0 } ^ { 3 } 2 \pi t ( 100 e ^ { - 0.55 t } )$$

Evaluate

$$600\pi te^{-\frac{11t}{20}}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2\times 100\pi te^{-\frac{11t}{20}}\mathrm{d}x$$

Find the integral of 2\times 100\pi te^{-\frac{11t}{20}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2\times 100\pi te^{-\frac{11t}{20}}x$$

Simplify.

$$200e^{-\frac{11t}{20}}\pi tx$$

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.

$$200e^{-0.55t}\pi t\times 3-200e^{-0.55t}\pi t\times 0$$

Simplify.

$$600e^{-\frac{11t}{20}}\pi t$$

Differentiate w.r.t. t

$$30\pi \left(20-11t\right)e^{-\frac{11t}{20}}$$