Topic: Engineering Mathematics
The value of $\iint_{0}^{1}\frac{dxdy}{\sqrt{(1-x^{2})(1-y^{2})}}is=$
- A. $\frac{\pi^{2}}{2}$
- B. $\frac{\pi}{4}$
- C. $\frac{\pi^{2}}{4}$
- D. $\frac{\pi}{2}$
Answer: C. $\frac{\pi^{2}}{4}$
Explanation: $I=\int_{0}^{1}\int_{0}^{1}\frac{dxdy}{\sqrt{1-x^{2}}\sqrt{1-y^{2}}}$. Since the limits are independent, we can separate the integrals: $I = (\int_{0}^{1}\frac{dx}{\sqrt{1-x^{2}}}) \times (\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}})$. The integral of $\frac{1}{\sqrt{1-x^{2}}}$ is $\sin^{-1}x$. Evaluating from 0 to 1, we get $\sin^{-1}(1) - \sin^{-1}(0) = \frac{\pi}{2} - 0 = \frac{\pi}{2}$. Thus, $I = \frac{\pi}{2} \times \frac{\pi}{2} = \frac{\pi^{2}}{4}$.