# Math Help - Integration

1. ## Integration

Could you help me with this?

By considering the integral of $e^{-yx^2}$ over a suitable region of $\mathbb{R}^2$, show that for $0,

$\int_{0}^{\infty} \frac{e^{-ax^2} - e^{-bx^2}}{x^2} dx = \sqrt{\pi} ( \sqrt{b} - \sqrt {a} )$

2. the hint was already given, we have that $\frac{e^{-ax^2}-e^{-bx^2}}{x^2}=\int_a^b e^{-tx^2}\,dt,$ now reverse integration order (justified by Tonelli's Theorem since the integrand is non negative, becuase of $b>a$) and use the fact that $\int_{x\ge0}e^{-x^2}\,dx=\frac{\sqrt\pi}2,$ hence, the result.