What is the integral from 0 to 2 of z(z-2)^2 ?

The integral from 0 to 2 of z(z-2)^2 is 4/3

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integral from 0 to 2 of z(z-2)^2

\int_{0}^{2} z (z - 2)^{2} d z

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

\int_{0}^{2} z (z - 2)^{2} d z

Expand

z (z - 2)^{2} : z^{3} - 4 z^{2} + 4 z

= \int_{0}^{2} z^{3} - 4 z^{2} + 4 z d z

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{2} z^{3} d z - \int_{0}^{2} 4 z^{2} d z + \int_{0}^{2} 4 z d z

\int_{0}^{2} z^{3} d z = 4

\int_{0}^{2} 4 z^{2} d z = \frac{32}{3}

\int_{0}^{2} 4 z d z = 8

= 4 - \frac{32}{3} + 8

Simplify

4 - \frac{32}{3} + 8 : \frac{4}{3}

= \frac{4}{3}

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What is the integral from 0 to 2 of z(z-2)^2 ?
The integral from 0 to 2 of z(z-2)^2 is 4/3