Question

Assertion :The area bounded by $y=(x-1{)}^2$, the y-axis and the line $y=2$ is ${\int }_2^2(2-x(x-1{)}^2)dx=\frac{10}{3}$. and Reason: The curve $y={x(x-1)}^2$ is intersected by $y=2$ at $x=2$ only and for $\displaystyle 0

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect, Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Solving $y={x(x-1)}^2$ and $y=2$ we get $x=2$
Hence $y={x(x-1)}^2$ intersects the line $y=2$ at $x=2$ only.
Curve lies below the line $y=2\forall x\in (0,2)$
hence required area is ${\int }_0^2\{2-{x(x-1)}^2\}dx$
$|2x-\frac{x^4}{4}-\frac{x^2}{2}+\frac{{2x}^3}{3}{|}_0^2=|4-4-2+\frac{16}{3}|=\frac{10}{3}$