What is the integral from 0 to 8 of 2piy(y^{2/3}) ?

The integral from 0 to 8 of 2piy(y^{2/3}) is 192pi

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integral from 0 to 8 of 2piy(y^{2/3})

\int_{0}^{8} 2 π y (y^{\frac{2}{3}}) d y

192 π

Decimal

603.18578 \dots

Solution steps

\int_{0}^{8} 2 π y y^{\frac{2}{3}} d y

Take the constant out:

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

= 2 π \cdot \int_{0}^{8} y y^{\frac{2}{3}} d y

Simplify

y y^{\frac{2}{3}} : y^{\frac{5}{3}}

= 2 π \cdot \int_{0}^{8} y^{\frac{5}{3}} d y

Apply the Power Rule

= 2 π [\frac{3}{8} y^{\frac{8}{3}}]_{0}^{8}

Compute the boundaries:

96

= 2 π 96

Simplify

= 192 π

What is the integral from 0 to 8 of 2piy(y^{2/3}) ?
The integral from 0 to 8 of 2piy(y^{2/3}) is 192pi