Question

The value of $\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2)} dx dy$ is}

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{6}$
• C. $\frac{\sqrt{\pi}}{2}$
• D. $\frac{\pi}{4}$

Hint:

The standard Gaussian integral $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$ is extremely useful.
By symmetry, over the interval $[0, \infty)$, it is $\frac{\sqrt{\pi}}{2}$.
The product of two such independent integrals is $(\frac{\sqrt{\pi}}{2})^2 = \frac{\pi}{4}$.

Answer 1

The Correct Option is D