JEE Maths - Application Of Integrals

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Question - 1

The area bounded by the x-axis, the curve y = (x) and the lines x = 1, X = b, is equal to \(\sqrt{\mathrm{b}^{2}+1}-\sqrt{2} \text { for all } \mathrm{b}>1\) then f(x) is _________

  • A \(\sqrt{x-1}\)
  • B \(\sqrt{x+1}\)
  • C \(\sqrt{x^{2}+1}\)
  • D \(\frac{x}{\sqrt{1+x^{2}}}\)

Question - 2

Area of region bounded by \([x]^{2}=[y]^{2}\) if x \(\in\) [1, 5] (where [.] represents the greatest integer function), is __________

  • A l0 sq. unit
  • B 8 sq. unit
  • C 6 sq. unit
  • D 5 sq. unit

Question - 3

If a point P moves such that its distance from line \(y=\sqrt{3} x-7\) is same as its distance from \((2 \sqrt{3},-1)\) ,then area of the curve, described by P, enclosed between the coordinate axes is ________

  • A \(\frac{\sqrt{3}}{2}\)
  • B \(2 \sqrt{3}\)
  • C 6
  • D \(\sqrt{3}\)

Question - 4

The area bounded by the curve \(f(x)=\| \tan x+\) \(\cot x|-| \tan x-\cot x \|\) between the lines x = 0, \(x=\frac{\pi}{2}\) and the x-axis, is _______

  • A \(\ln \sqrt{2}\)
  • B \(2 \ln 2\)
  • C \(\ln 4\)
  • D \(\sqrt{2} \ln 2\)

Question - 5

If the lines y = mx + 2 the parabola \(2 y=x^{2}\) at points \(\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right) \text { where }\left(x_{1} then the value of m for which \(\int_{x_{1}}^{x_{2}}\left(m x+2-\frac{x^{2}}{2}\right) d x\) minimum is ___________

  • A \(\sqrt{2}\)
  • B \(\frac{8}{3}\)
  • C \(\frac{1}{\sqrt{3}}\)
  • D 0

Question - 6

The area bounded by the curve f(x) = r + sin x and its inverse function between the ordinates x = 0 to x = \(2 \pi\) is ___________

  • A 4\(\pi\) square units
  • B 8\(\pi\) square units
  • C 4 square units
  • D 2 square units

Question - 7

The area bounded by the curve \(y^{2}(2 a-x)=x^{3}\) and the line x = 2a is __________

  • A \(3 \pi \mathrm{a}^{2} \text { sq. unit }\)
  • B \(\frac{3 \pi \mathrm{a}^{2}}{2} \text { sq. unit }\)
  • C \(\frac{3 \pi \mathrm{a}^{2}}{4} \text { sq. unit }\)
  • D \(\frac{6 \pi \mathrm{a}^{2}}{5} \text { sq. unit }\)

Question - 8

The area bounded by the curves \(y=x e^{x}\), \(y=x e^{-x}\) and the line x = 1 is ____________

  • A \(\frac{2}{e}\)
  • B \(1-\frac{2}{e}\)
  • C \(\frac{1}{e}\)
  • D \(1-\frac{1}{e}\)

Question - 9

Area enclosed by the curve \(|x+y-1|+|2 x+y-1|=1\) is _________

  • A 2 sq. unit
  • B 3 sq. units
  • C 6 sq. units
  • D 7 sq. units

Question - 10

The area bounded by the curve \(f(x)=\max \cdot\left\{4-x^{2},|x-2|,(x-2)^{1 / 3}\right\}\) \(\text { for } x \in[-2,4]\) and x-axis is __________

  • A \(\frac{45}{.4}\) sq. units
  • B \(\frac{23}{4}\)  sq. units
  • C \(\frac{59}{4}\)  sq. units
  • D \(\frac{53}{4}\)  sq. units
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