Large areas!

Calculus Level 5

What is the area bounded by the curve $y=x^6( \pi - x) ^8$ between $x=0$ to $x=\pi$ and $x$ -axis?

The area can be represented as $\dfrac{ \pi ^ab! c!}{d!}$ , where $a$ , $b$ , $c$ , and $d$ are positive integers. Find the value of $a+b+c+d$ .

Notation: $!$ is the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

The answer is 44.

1 solution

Chew-Seong Cheong
Dec 20, 2016

$\begin{aligned} A & = \int_0^\pi x^6(\pi - x)^8 \ dx & \small \color{#3D99F6} \text{Let } u = \frac x \pi, \ dx = \pi \ du \\ & = \int_0^1 (\pi u)^6 \pi^8 (1 - u)^8 \pi \ du \\ & = \pi^{15} \color{#3D99F6} \int_0^1 u^6 (1 - u)^8 \ du \\ & = \pi^{15} \color{#3D99F6} B(7,9) & \small \color{#3D99F6} B(m,n) \text{ is beta function.} \\ & = \pi^{15} \frac {\Gamma(7)\Gamma(9)}{\Gamma(16)} & \small \color{#3D99F6} \Gamma(n) \text{ is gamma function.} \\ & = \frac {\pi^{15}6!8!}{15!} \end{aligned}$

$\implies a+b+c+d = 15+6+8+15 = \boxed{44}$

References:

Beta function
Gamma function

Large areas!

The answer is 44.

1 solution

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