What is the integral from 0 to 4 of sqrt(4x)-x ?

The integral from 0 to 4 of sqrt(4x)-x is 8/3

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integral from 0 to 4 of sqrt(4x)-x

\int_{0}^{4} 4 x - x d x

\frac{8}{3}

Decimal

2.66666 \dots

Solution steps

\int_{0}^{4} 4 x - x d x

4 x = 2 x

= \int_{0}^{4} 2 x - x d x

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}^{4} 2 x d x - \int_{0}^{4} x d x

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

\int_{0}^{4} x d x = 8

= \frac{32}{3} - 8

Simplify

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

= \frac{8}{3}

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What is the integral from 0 to 4 of sqrt(4x)-x ?
The integral from 0 to 4 of sqrt(4x)-x is 8/3