JEE Advanced 2024 Paper 2 · Q17 · Definite Integrals

Question

PARAGRAPH "II": Let $f : \left[0, \dfrac{\pi}{2}\right] \to [0, 1]$ be the function defined by $f(x) = \sin^{2} x$ and let $g : \left[0, \dfrac{\pi}{2}\right] \to [0, \infty)$ be the function defined by $g(x) = \sqrt{\dfrac{\pi x}{2} - x^{2}}$.

The value of $\displaystyle \frac{16}{\pi^{3}} \int_{0}^{\pi/2} f(x)\,g(x)\,dx$ is ______.

SolveKar AI · Accepted Answer

From Q16, $2\int_{0}^{\pi/2} f(x) g(x)\,dx = \int_{0}^{\pi/2} g(x)\,dx$. The integral $\int_{0}^{\pi/2} g(x)\,dx$ is the area of a semicircle of radius $\pi/4$, which equals $\dfrac{1}{2}\pi(\pi/4)^{2} = \dfrac{\pi^{3}}{32}$. Hence $\int_{0}^{\pi/2} f(x) g(x)\,dx = \dfrac{\pi^{3}}{64}$. Therefore $\dfrac{16}{\pi^{3}} \cdot \dfrac{\pi^{3}}{64} = \dfrac{16}{64} = 0.25$.

JEE Advanced 2024 Paper 2 · Q17 · Definite Integrals

Solution