First slide

Area of bounded Regions

Question

$Find the area, lying above the x -axis and included between the circle x^{2} + y^{2} = 8 x and the parabola y^{2} = 4 x$

Moderate

A

$\frac{32}{3} + 4 π s q u n i t s$
B

$\frac{16}{3} + 4 π s q u n i t s$
C

$\frac{32}{3} + 16 π s q u n i t s$
D

$\frac{32}{3} + 8 π s q u n i t s$

Solution

$Solving the curves, we get x^{2} + 4 x = 8 x \Rightarrow x = 0, 4$
$\begin{matrix} = \int_{0}^{4} y_{parabola} d x + \int_{4}^{8} y_{circle} d x \\ A = \int_{0}^{4} 2 \sqrt{x} d x + 4 π \\ = \frac{4}{3} {[x^{3 / 2}]}_{0}^{4} + 4 π \\ = \frac{4}{3} \times 4 \sqrt{4} + 4 π \\ = \frac{32}{3} + 4 π square units \end{matrix}$

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