Question

please help! I'm trying to find a constant function (g(x)=constant) on the interval [0, pi^1/3] so that the area under the graph of g is the same as the area under the graph of f(x)=xsin(x^3) (on the same interval). All I'm sure about is that g(x) is a horizontal line...any help is appreciated.

Answer 1

This is the graph of the function f(x):

Answer 2

The first thing to do is determine what the area under \(f(x)\) is on the interval:
\[\int_0^{\pi^{1/3}}x\sin x^3~dx\]
The best method I can think to use here is a power series representation for sine:
\[\begin{align*}\sin x=\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}&~~\Rightarrow~~\sin x^3=\sum_{n=0}^\infty \frac{(-1)^nx^{6n+3}}{(2n+1)!}\\&~~\Rightarrow~~x\sin x^3=\sum_{n=0}^\infty \frac{(-1)^nx^{6n+4}}{(2n+1)!}\end{align*}\]
Listing the first few terms:
\[x\sin x^3=x^4-\frac{x^{10}}{3!}+\frac{x^{16}}{5!}-\frac{x^{22}}{7!}+\cdots\]
Try integrating from there.

Answer 3

The attached Mathematica solution calculated the area to 40 numerical digits.

Answer 4

Thank you both! I didn't realize how simple the question was...I think the constant function is g(x)=0.317215937 (the area under this line is equal to the area under f(x)).