First slide

Area of bounded Regions

Question

$The area of the region enclosed between the curves x = y^{2} - 1 and x = | y | \sqrt{1 - y^{2}} is$ $- - - - - - sq. units$

Difficult

A

2
B

$\frac{4}{3}$
C

1
D

$\frac{1}{8}$

Solution

$\begin{array}{l} Required area A = 2 \int_{0}^{1} (y \sqrt{1 - y^{2}} - (y^{2} - 1)) d y \\ = - \int_{0}^{1} {(1 - y^{2})}^{\frac{1}{2}} (- 2 y) d y - 2 \int_{0}^{1} y^{2} d y + 2 \int_{0}^{1} d y \\ = - \frac{2}{3} {({(1 - y^{2})}^{\frac{3}{2}})}^{1}_{0} - 2 {(\frac{y^{3}}{3})}_{0}^{1} + 2 (y)_{0}^{1} \\ = \frac{2}{3} - \frac{2}{3} + 2 \\ = 2 sq.units \end{array}$

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