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Application of Integration question

2008 · Shift 1 · Q30
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  5. /2008 · Shift 1 · Q30

Application of Integration question

2008 · Shift 1 · Q30

JEE AdvancedMathematicsApplication of IntegrationMCQ+3 / −1
Consider the functions defined implicitly by the equation y3−3y+x=0y^3-3y+x=0y3−3y+x=0 on various intervals in the real line. If x∈(−∞,−2)∪(2,∞)x\in(-\infty,-2)\cup(2,\infty)x∈(−∞,−2)∪(2,∞), the equation implicitly defines a unique real valued differentiable function y=f(x)y=f(x)y=f(x). If x∈(−2,2)x\in(-2,2)x∈(−2,2), the equation implicitly defines a unique real valued differentiable function y=g(x)y=g(x)y=g(x) satisfying g(0)=0g(0)=0g(0)=0The area of the region bounded by the curve y=f(x),y=f(x),y=f(x), the xxx-axis, and the lines x=ax=ax=a and x=bx=bx=b, where −∞<a<b<−2,- \infty \lt a \lt b \lt - 2,−∞<a<b<−2, is :
  1. A
    ∫abx3((f(x))2−1)dx+bf(b)−af(a)\int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx + bf\left( b \right) - af\left( a \right)a∫b​3((f(x))2−1)x​dx+bf(b)−af(a)
  2. B
    −∫abx3((f(x))2−1)dx+bf(b)−af(a)- \int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx + bf\left( b \right) - af\left( a \right)−a∫b​3((f(x))2−1)x​dx+bf(b)−af(a)
  3. C
    ∫abx3((f(x))2−1)dx−bf(b)+af(a)\int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx - bf\left( b \right) + af\left( a \right)a∫b​3((f(x))2−1)x​dx−bf(b)+af(a)
  4. D
    −∫abx3((f(x))2−1)dx−bf(b)+af(a)- \int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx - bf\left( b \right) + af\left( a \right)−a∫b​3((f(x))2−1)x​dx−bf(b)+af(a)
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