Area Under Curves Question 4
Question 4 - 29 January - Shift 1
Let $\Delta$ be the area of the region $\lbrace (x, y) \in R^{2}: x^{2}+y^{2} \leq 21, y^{2} \leq 4 x, x \geq 1\rbrace $. Then $\frac{1}{2}(\Delta-21 \sin ^{-1} \frac{2}{\sqrt{7}})$ is equal to
(1) $2 \sqrt{3}-\frac{1}{3}$
(2) $\sqrt{3}-\frac{2}{3}$
(3) $2 \sqrt{3}-\frac{2}{3}$
(4) $\sqrt{3}-\frac{4}{3}$
Show Answer
Answer: (4)
Solution:
Formula: Area between two curves - Area bounded by two intersecting curves and lines parallel to $y-$ axis
Area $2 \int_1^{3} 2 \sqrt{x} d x+2 \int_3^{\sqrt{21}} \sqrt{21-x^{2} d x}$
$\Delta=\frac{8}{3}(3 \sqrt{3}-1)+21 \sin ^{-1}(\frac{2}{\sqrt{7}})-6 \sqrt{3}$
$\frac{1}{2}(\Delta-21 \sin ^{-1}(\frac{2}{\sqrt{7}}))=\frac{2 \sqrt{3}-\frac{8}{3}}{2}$
$=\sqrt{3}-\frac{4}{3}$
