What is the integral from 0 to 3 of sqrt(x(3-x)) ?

The integral from 0 to 3 of sqrt(x(3-x)) is (9pi)/8

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integral from 0 to 3 of sqrt(x(3-x))

\int_{0}^{3} x (3 - x) d x

\frac{9 π}{8}

Decimal Notation

3.53429 \dots

Solution steps

\int_{0}^{3} x (3 - x) d x

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

x (3 - x) = x (3 - x)

= \int_{0}^{3} x 3 - x d x

Apply Integration By Parts

: [\frac{2}{3} x^{\frac{3}{2}} 3 - x - \int - \frac{x ^{\frac{3}{2}}}{3 3 - x} d x]_{0}^{3}

= [\frac{2}{3} x^{\frac{3}{2}} 3 - x - \int - \frac{x ^{\frac{3}{2}}}{3 3 - x} d x]_{0}^{3}

\int - \frac{x ^{\frac{3}{2}}}{3 3 - x} d x = \frac{1}{48} (32 x^{\frac{3}{2}} - x + 3 + 27 (4 arcsin (\frac{- x + 3}{3}) - sin (4 arcsin (\frac{- x + 3}{3}))))

= [\frac{2}{3} x^{\frac{3}{2}} 3 - x - \frac{1}{48} (32 x^{\frac{3}{2}} - x + 3 + 27 (4 arcsin (\frac{- x + 3}{3}) - sin (4 arcsin (\frac{- x + 3}{3}))))]_{0}^{3}

Simplify

[\frac{2}{3} x^{\frac{3}{2}} 3 - x - \frac{1}{48} (32 x^{\frac{3}{2}} - x + 3 + 27 (4 arcsin (\frac{- x + 3}{3}) - sin (4 arcsin (\frac{- x + 3}{3}))))]_{0}^{3} : [\frac{9}{16} (sin (4 arcsin (\frac{- x + 3}{3})) - 4 arcsin (\frac{- x + 3}{3}))]_{0}^{3}

= [\frac{9}{16} (sin (4 arcsin (\frac{- x + 3}{3})) - 4 arcsin (\frac{- x + 3}{3}))]_{0}^{3}

Compute the boundaries:

\frac{9 π}{8}

= \frac{9 π}{8}

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What is the integral from 0 to 3 of sqrt(x(3-x)) ?
The integral from 0 to 3 of sqrt(x(3-x)) is (9pi)/8