What is the integral from 0 to b of (a-asqrt(x/b)) ?

The integral from 0 to b of (a-asqrt(x/b)) is a b/3

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integral from 0 to b of (a-asqrt(x/b))

\int_{0}^{b} (a - a \frac{x}{b}) d x

a \frac{b}{3}

Solution steps

\int_{0}^{b} a - a \frac{x}{b} 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}^{b} a d x - \int_{0}^{b} a \frac{x}{b} d x

\int_{0}^{b} a d x = ab

\int_{0}^{b} a \frac{x}{b} d x = a (b - \frac{b}{3})

= ab - a (b - \frac{b}{3})

Simplify

ab - a (b - \frac{b}{3}) : a \frac{b}{3}

= a \frac{b}{3}

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What is the integral from 0 to b of (a-asqrt(x/b)) ?
The integral from 0 to b of (a-asqrt(x/b)) is a b/3