Area Under Curves Question 9

Question 9 - 30 January - Shift 2

Let $A$ be the area of the region

${(x, y): y \geq x^{2}, y \geq(1-x)^{2}, y \leq 2 x(1-x)}$

Then $540 A$ is equal to

Show Answer

Answer: (25)

Solution:

Formula: Area between two curves - Area enclosed between two curves intersecting at two different points

$A=2 \int _{\frac{1}{3}}^{\frac{1}{2}}(2 x-2 x^{2}-(1-x)^{2}) d x$

$A=2[2 x^{2}-x^{3}-x] _{1 / 3}^{1 / 2}$

$\therefore A=\frac{5}{108} \Rightarrow 540 A=\frac{5}{108} \times 540=25$