Area Ques 6
- The area of the region bounded by the curve $y=f(x)$, the $X$-axis and the lines $x=a$ and $x=b$, where $-\infty<a<b<-2$, is
(a) $\int_a^b \frac{x}{3\left[\{f(x)\}^2-1\right]} d x+b f(b)-a f(a)$
(b) $-\int_a^b \frac{x}{3\left[\{f(x)\}^2-1\right]} d x+b f(b)-a f(a)$
(c) $\int_a^b \frac{x}{3\left[\{f(x)\}^2-1\right]} d x-b f(b)+a f(a)$
(d) $-\int_a^b \frac{x}{\left.3[f f(x)\}^2-1\right]} d x-b f(b)+a f(a)$
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Answer:
Correct Answer: 6.(a)
Solution: (a)
$ \begin{aligned} & \text { Required area }=\int_a^b y d x=\int_a^b f(x) d x \\ & =[f(x) \cdot x]_a^b-\int_a^b f^{\prime}(x) x d x \\ & =b f(b)-a f(a)-\int_a^b f^{\prime}(x) x d x \\ & =b f(b)-a f(a)+\int_a^b \frac{x d x}{3\left[\{f(x)\}^2-1\right]} \\ & {\left[\because f^{\prime}(x)=\frac{d y}{d x}=\frac{-1}{3\left(y^2-1\right)}=\frac{-1}{3\left[\{f(x)\}^2-1\right]}\right]} \\ \end{aligned} $
BETA
