Definition of integrability

Question

By the definition of Riemann integrals I have to show that: $\int_{-a}^{a} f(x^2) dx=2\int_{0}^{a} f(x^2) dx$ (given that both integrals exist)

I thought to approach it as: $\int_{-a}^{a} f(x^2) dx=\int_{-a}^{0} f(x^2) dx+\int_{0}^{a} f(x^2) dx$. And somehow to show that integrals are equal to each other on the right side, but I am not sure how to approach the problem.

Answer 1

Note that $f(x^2)$ is even since $f((-x)^2)=f(x^2)$. So$$\begin{align}\int_{-a}^af(x^2)\text{d}x&=\int_{-a}^0f(x^2)\text{d}x+\int_0^af(x^2)\text{d}x\\&=\int_0^af(x^2)\text{d}x+\int_0^af(x^2)\text{d}x\\&=2\int_0^af(x^2)\text{d}x.\end{align}$$

Answer 2

If $\{x_0,x_1,...,x_n\}$ is a partition of $[0,a]$ then $\{-x_n,-x_{n-1},...,-x_1,x_0,x_1,...,x_n\}$ is a partition of $[-a,a]$. Just write down the upper and lower sums for this partition and take limit as the norm of the partition tends to $0$.