What is the integral from 2 to 6 of pi(2/y)^2 ?

The integral from 2 to 6 of pi(2/y)^2 is (4pi)/3

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integral from 2 to 6 of pi(2/y)^2

\int_{2}^{6} π (\frac{2}{y})^{2} d y

\frac{4 π}{3}

Decimal

4.18879 \dots

Solution steps

\int_{2}^{6} π (\frac{2}{y})^{2} d y

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= π \cdot \int_{2}^{6} (\frac{2}{y})^{2} d y

Apply Integration By Parts

: [\frac{4}{y} - \int - \frac{8}{y ^{2}} d y]_{2}^{6}

= π [\frac{4}{y} - \int - \frac{8}{y ^{2}} d y]_{2}^{6}

\int - \frac{8}{y ^{2}} d y = \frac{8}{y}

= π [\frac{4}{y} - \frac{8}{y}]_{2}^{6}

Simplify

π [\frac{4}{y} - \frac{8}{y}]_{2}^{6} : π [- \frac{4}{y}]_{2}^{6}

= π [- \frac{4}{y}]_{2}^{6}

Compute the boundaries:

\frac{4}{3}

= π \frac{4}{3}

Simplify

= \frac{4 π}{3}

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What is the integral from 2 to 6 of pi(2/y)^2 ?
The integral from 2 to 6 of pi(2/y)^2 is (4pi)/3