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JEE Advanced 2008Paper 1
Exam
JEE Advanced
Year
2008
Paper
Paper 1
Subject
Mathematics
Chapter
Area Under Curves

JEE Advanced 2008 · Paper 1 · Mathematics · Area Under Curves

Paragraph: Consider the functions defined implicitly by the...

Paragraph: Consider the functions defined implicitly by the equation y3−3y+x=0y^3-3 y+x=0 on various intervals in the real line. If x∈(−∞,−2)∪(2,∞)x \in(-\infty,-2) \cup(2, \infty), the equation implicitly defines a unique real valued differentiable function y=f(x)y=f(x). If x∈(−2,2)x \in(-2,2), the equation implicitly defines a unique real valued differentiable function y=g(x)y=g(x), satisfying g(0)=0g(0)=0.Question: The area of the region bounded by the curve y=f(x)y=f(x), the XX-axis and the lines x=ax=a and x=bx=b, where −∞<a<b<−2-\infty < a < b < -2, is

Options

  1. A∫abx3[f(x)2−1]dx+bf(b)−af(a)\int_a^b \frac{x}{3\left[f(x)^2-1\right]} d x+b f(b)-a f(a)
  2. B−∫abx3[(f(x))2−1]dx+bf(b)−af(a)-\int_a^b \frac{x}{3\left[(f(x))^2-1\right]} d x+b f(b)-a f(a)
  3. C∫abx3[(f(x))2−1]dx−bf(b)+af(a)\int_a^b \frac{x}{3\left[(f(x))^2-1\right]} d x-b f(b)+a f(a)
  4. D−∫abx3[f(x)2−1]dx−bf(b)−af(a)-\int_a^b \frac{x}{3\left[f(x)^2-1\right]} d x-b f(b)-a f(a)

Correct answer

A. ∫abx3[f(x)2−1]dx+bf(b)−af(a)\int_a^b \frac{x}{3\left[f(x)^2-1\right]} d x+b f(b)-a f(a)

Step-by-step solution

Required area =∫abydx=∫abf(x)dx=\int_a^b y d x=\int_a^b f(x) d x =[f(x) ⋅\cdotx]_a^b-\int_a^b f^{\prime}(x) ⋅\cdotx d x=b f(b)-a f(a)-\int_a^b f^{\prime}(x) ⋅\cdotx d x =b f(b)-a f(a)+\int_a^b x3[(f(x))2−1]\frac{x}{3\left[(f(x))^2-1\right]}d x As, f′(x)=dydx=−13(y2−1)=−13[(f(x))2−1]\quad 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]}
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