Question

Area bounded by the curve 2y² = 3x and the line x+y = 3 outside the circle (x-3)² + y² = 2 and above the x-axis is A. The value of 4(π +4A) is?

Answer 1

The correct answer is : 42

A = required area
\(=\int\limits^{\frac{3}{2}}_0\left[(3-y)-(\frac{2y^2}{3})\right]dy-\pi(\sqrt2)^2.\frac{1}{8}\)
\(⇒\left(3y-\frac{y^2}{2}-\frac{2}{9}y^3\right)\big|^{\frac{3}{2}}_{0}-\frac{\pi}{4}\)
\(⇒3.\frac{3}{2}-\frac{9}{8}-\frac{2}{9}.\frac{27}{8}-\frac{\pi}{4}\)
\(⇒\frac{36-9-6}{8}-\frac{\pi}{4}\)
\(=\frac{21}{8}-\frac{\pi}{4}\)
\(⇒4(\pi+4A)=4(\frac{21}{2})\)
\(=42\)