Solve ∫ (from 3 to 5) of (3/y)2piy wrt y

$$\int _ { 3 } ^ { 5 } ( \frac { 3 } { y } ) 2 \pi y d y$$

Evaluate

$$12\pi \approx 37.699111843$$

Solution Steps

Express \frac{3}{y}\times 2 as a single fraction.

$$\int _{3}^{5}\frac{3\times 2}{y}\pi y\mathrm{d}y$$

Express \frac{3\times 2}{y}\pi as a single fraction.

$$\int _{3}^{5}\frac{3\times 2\pi }{y}y\mathrm{d}y$$

Cancel out y and y.

$$\int _{3}^{5}3\times 2\pi \mathrm{d}y$$

Multiply 3 and 2 to get 6.

$$\int _{3}^{5}6\pi \mathrm{d}y$$

Evaluate the indefinite integral first.

$$\int 6\pi \mathrm{d}y$$

Find the integral of 6\pi using the table of common integrals rule \int a\mathrm{d}y=ay.

$$6\pi y$$

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.

$$6\pi \times 5-6\pi \times 3$$

Simplify.

$$12\pi$$