Let f : [-2,3][0,) be a continuous function such that f(1-x)=f(x) for all x[-2,3].

Question

Let $f : [-2,3]	o[0,\infty)$ be a continuous function such that $f(1-x)=f(x)$ for all $x\in[-2,3]$. If $R_1$ is the area of the region bounded by $y=f(x)$, $x=-2$, $x=3$, and the $x$-axis, and $R_2 = \displaystyle\int_{-2}^3 xf(x)\,dx$, then:

Accepted Answer

Using the property $\displaystyle\int_a^b g(x)\,dx=\int_a^b g(a+b-x)\,dx$ with $a=-2,b=3$ (so $a+b=1$):

$$R_2 = \int_{-2}^3(1-x)f(1-x)\,dx = \int_{-2}^3(1-x)f(x)\,dx$$

Adding this to the original definition of $R_2$:

$$2R_2 = \int_{-2}^3 xf(x)\,dx + \int_{-2}^3(1-x)f(x)\,dx = \int_{-2}^3 f(x)\,dx = R_1$$

So $R_1 = 2R_2$.

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