Allied & ICAR Exam Bank • ALLIED_1026NAT • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Multiple integrals
Question Statement:
The value of the double integral $\int_0^1 \int_0^{\sqrt{1-x^2}} \left(x^2 + y^2\right) dy dx$ is $\frac{\pi}{k}$. The integer value of $k$ is:
Answer Key:
8
Step-by-Step Solution & Explanation
Converting to polar coordinates $(r, \theta)$: the region is the first quadrant of the unit circle $r \in [0, 1], \theta \in [0, \pi/2]$. The integrand $x^2 + y^2 = r^2$, and $dA = r dr d\theta$. $\int_0^{\pi/2} d\theta \int_0^1 r^3 dr = \frac{\pi}{2} \left[\frac{r^4}{4}\right]_0^1 = \frac{\pi}{2} \times \frac{1}{4} = \frac{\pi}{8}$. Thus $k = 8$.