What is the integral from 0 to 4 of (sqrt(x)-1/2 x) ?

The integral from 0 to 4 of (sqrt(x)-1/2 x) is 4/3

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integral from 0 to 4 of (sqrt(x)-1/2 x)

\int_{0}^{4} (x - \frac{1}{2} x) d x

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

\int_{0}^{4} x - \frac{1}{2} 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} x d x - \int_{0}^{4} \frac{1}{2} x d x

\int_{0}^{4} x d x = \frac{16}{3}

\int_{0}^{4} \frac{1}{2} x d x = 4

= \frac{16}{3} - 4

Simplify

\frac{16}{3} - 4 : \frac{4}{3}

= \frac{4}{3}

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What is the integral from 0 to 4 of (sqrt(x)-1/2 x) ?
The integral from 0 to 4 of (sqrt(x)-1/2 x) is 4/3