First slide

Area of bounded Regions

Question

$If the area enclosed by curve y = f (x) and y = x^{2} + 2 between the abscissa x = 2 and x = α, α > 2, is$
$(α^{3} - 4 α^{2} + 8) sq. unit. It is known that curve y = f (x) lies below the parabola y = x^{2} + 2$

Moderate

A

1
B

2
C

3
D

4

Solution

According to the question,
$α^{3} - 4 α^{2} + 8 = \int_{2}^{α} (x^{2} + 2 - f (x)) d x$
$Differentiating w.r.t. α on both sides, we get$
$\begin{aligned} 3 α^{2} - 8 α = α^{2} + 2 - f (α) \\ ∴ f (x) = - 2 x^{2} + 8 x + 2 \end{aligned}$

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