# Evaluate the integral: {eq}\displaystyle \int_0^3 \sqrt {6 x - x^2} \ dx {/eq}.

## Question:

Evaluate the integral:

{eq}\displaystyle \int_0^3 \sqrt {6 x - x^2} \ dx {/eq}.

## Working rule in Definite Integration:

Let function {eq}m {/eq} is continous on {eq}[p,q] {/eq} and {eq}M {/eq} is a differentiable function on {eq}(p,q) {/eq} such that, {eq}\forall \,x \in (p,q),\,\frac{d}{{dx}}M(x) = m(x) {/eq} then,

{eq}\int\limits_p^q {m(x)dx = M(p) - M(q)} {/eq}.

The integrated function must be evaluated in both the extremes of the interval and subtract their values.

Given that: {eq}\displaystyle \int\limits_0^3 {\sqrt {6x - {x^2}} dx} {/eq}

{eq}\displaystyle\ \eqalign{ & \int\limits_0^3 {\sqrt {6x - {x^2}} dx} = \int\limits_0^3 {\sqrt {9 - 9 + 6x - {x^2}} dx} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \int\limits_0^3 {\sqrt {9 - {{(x - 3)}^2}} dx} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{Factor: }}9 - {{(x - 3)}^2} = 9 - 9 + 6x - {x^2}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \int\limits_0^3 {\sqrt {{3^2} - {{(x - 3)}^2}} dx} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left[ {\frac{{x - 3}}{2}\sqrt {{3^2} - {{(x - 3)}^2}} + \frac{{{3^2}}}{2}{{\sin }^{ - 1}}\left( {\frac{{x - 3}}{3}} \right)} \right]_0^3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\int {\sqrt {{a^2} - {x^2}} dx = \frac{x}{2}\sqrt {{a^2} - {x^2}} + \frac{{{a^2}}}{2}{{\sin }^{ - 1}}\left( {\frac{x}{a}} \right) + c} } \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left[ {\left( {\frac{{3 - 3}}{2}\sqrt {{3^2} - {{(3 - 3)}^2}} + \frac{{{3^2}}}{2}{{\sin }^{ - 1}}\left( {\frac{{3 - 3}}{3}} \right)} \right) - \left( {\frac{{0 - 3}}{2}\sqrt {{3^2} - {{(0 - 3)}^2}} + \frac{{{3^2}}}{2}{{\sin }^{ - 1}}\left( {\frac{{0 - 3}}{3}} \right)} \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left[ {0 - \left( {0 + \frac{9}{2}{{\sin }^{ - 1}}\left( { - 1} \right)} \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{9}{2}{\sin ^{ - 1}}\left( 1 \right) \cr & \int\limits_0^3 {\sqrt {6x - {x^2}} dx} = \frac{{9\pi }}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\sin \frac{\pi }{2} = 1} \right) \cr} {/eq}