JEE Main & Advanced Mathematics Definite Integration Question Bank

done Self Evaluation Test - Application of Integrals Total Questions - 50

• question_answer1)
  If the area enclosed by \[{{y}^{2}}=4\,ax\text{ }is\text{ }\frac{1}{3}sq.\] unit, then the roots of the equation \[{{x}^{2}}+2x=a,\] are
  A)
  -4 and 2 done clear
  B)
  2 and 4 done clear
  C)
  -2 and -4 done clear
  D)
  8 and -8 done clear
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• question_answer2)
  Let f(x) be a continuous function such that the area bounded by the curve y = f(x), x-axis and the lines x = 0 and x = a is \[\frac{{{a}^{2}}}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos \] a, then \[f\left( \frac{\pi }{2} \right)=\]
  A)
  1 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[\frac{1}{3}\] done clear
  D)
  None of these done clear
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• question_answer3)
   The area bounded by \[y={{x}^{2}}+3\] and \[y=2x+3\] is (in sq. units)
  A)
  \[\frac{12}{7}\] done clear
  B)
  \[\frac{4}{3}\] done clear
  C)
  \[\frac{3}{4}\] done clear
  D)
  \[\frac{8}{3}\] done clear
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• question_answer4)
  The area of the smaller segment cut off from the circle \[{{x}^{2}}+{{y}^{2}}=9\,\,by\,\,x=1\] is
  A)
  \[\frac{1}{2}(9\,se{{c}^{-1}}3-\sqrt{8})\] sq. unit done clear
  B)
  \[(9\,se{{c}^{-1}}3-\sqrt{8})\] sq. unit done clear
  C)
  \[(\sqrt{8}-9\,\,se{{c}^{-1}}3)\]sq. unit done clear
  D)
  None of the above done clear
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• question_answer5)
  What is the area of the parabola \[{{x}^{2}}=y\] bounded by the line y = 1?
  A)
  \[\frac{1}{3}\] square unit done clear
  B)
  \[\frac{2}{3}\] square unit done clear
  C)
  \[\frac{4}{3}\] square units done clear
  D)
  2 square units done clear
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• question_answer6)
  What is the area under the curve \[y=\left| x \right|+\left| x-1 \right|\]between \[x=0\] and\[x=1\]?
  A)
  \[\frac{1}{2}\] done clear
  B)
  1 done clear
  C)
  \[\frac{3}{2}\] done clear
  D)
  2 done clear
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• question_answer7)
  What is the area of the region enclosed by \[~y=2\left| x \right|\] and\[y=4\]?
  A)
  2 square unit done clear
  B)
  4 square unit done clear
  C)
  8 square unit done clear
  D)
  16 square unit done clear
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• question_answer8)
  The triangle formed by the tangent to the curve \[f(x)={{x}^{2}}+bx-b\] at the point (1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of b is
  A)
  -1 done clear
  B)
  3 done clear
  C)
  -3 done clear
  D)
  1 done clear
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• question_answer9)
  Area bounded by the curves \[y=\left[ \frac{{{x}^{2}}}{64}+2 \right]([\cdot ]\] denotes the greatest integer function), \[v=x-1\] and \[x=0\], above the x-axis is
  A)
  2 sq. unit done clear
  B)
  3 sq. unit done clear
  C)
  4 sq. unit done clear
  D)
  None of these done clear
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• question_answer10)
  The area of the region (in sq. units), in the first quadrant bounded by the parabola \[y=9{{x}^{2}}\] and the lines \[x=0,\,\,y=1\] and \[y=4\], is:
  A)
  7/9 done clear
  B)
  14/3 done clear
  C)
  7/3 done clear
  D)
  14/9 done clear
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• question_answer11)
  The area bounded by the curve \[{{y}^{2}}(2a-x)={{x}^{3}}\]and the line \[x=2a\] is
  A)
  \[3\pi {{a}^{2}}\] sq. unit done clear
  B)
  \[\frac{3\pi {{a}^{2}}}{2}\] sq. unit done clear
  C)
   \[\frac{3\pi {{a}^{2}}}{4}\] sq. unit done clear
  D)
  \[\frac{6\pi {{a}^{2}}}{5}\] sq. unit done clear
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• question_answer12)
  The area enclosed between the curves \[y={{\log }_{e}}(x+e),x={{\log }_{e}}\left( \frac{1}{y} \right)\] and the x-axis is
  A)
  2 sq. units done clear
  B)
  1 sq. units done clear
  C)
  4 sq. units done clear
  D)
  None of these done clear
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• question_answer13)
  The value of a (a > 0) for which the area bounded by the curves \[y=\frac{x}{6}+\frac{1}{{{x}^{2}}},y=0,x=a\] and \[x=2a\] has the least value is
  A)
  2 done clear
  B)
  \[\sqrt{2}\] done clear
  C)
  \[{{2}^{1/3}}\] done clear
  D)
  1 done clear
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• question_answer14)
  The value of a (a > 0) for which the area bounded  by the curves \[y=\frac{x}{6}+\frac{1}{{{x}^{2}}},y=0,x=a\] and \[x=2a\]has the least value is
  A)
  2 done clear
  B)
  (b \[\sqrt{2}\] done clear
  C)
  \[{{2}^{1/3}}\] done clear
  D)
  1 done clear
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• question_answer15)
  If \[f(x)=a+bx+c{{x}^{2}}\], where \[c>0\] and \[{{b}^{2}}-4ac<0\], then the area enclosed by the coordinate axes, the line \[x=2\] and the curve \[y=f(x)\] is given by
  A)
  \[\frac{1}{3}\{4f(1)+f(2)\}\] done clear
  B)
         \[\frac{1}{2}\{f(0)+4f(1)+f(2)\}\] done clear
  C)
  \[\frac{1}{2}\{f(0)+4f(1)\}\] done clear
  D)
  \[\frac{1}{3}\{f(0)+4f(1)+f(2)\}\] done clear
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• question_answer16)
  The area enclosed between the curve \[y=lo{{g}_{e}}\left( x+e \right)\] and the coordinate axes is
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
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• question_answer17)
  The area of the figure bounded by \[{{y}^{2}}=2x+1\]and \[x-y=1\] is
  A)
  \[\frac{2}{3}\] done clear
  B)
  \[\frac{4}{3}\] done clear
  C)
  \[\frac{8}{3}\] done clear
  D)
  \[\frac{16}{3}\] done clear
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• question_answer18)
  The area bounded by the curve \[y=f(x),y=x\]and the lines \[x=1,x=t\] is \[(t+\sqrt{1+{{t}^{2}}})-\sqrt{2}-1\]sq. unit, for all t > 1. If f(x) satisfying f(x)>x for all x>1, then f(x) is equal to
  A)
  \[x+1+\frac{x}{\sqrt{1+{{x}^{2}}}}\] done clear
  B)
  \[x+\frac{x}{\sqrt{1+{{x}^{2}}}}\] done clear
  C)
  \[1+\frac{x}{\sqrt{1+{{x}^{2}}}}\] done clear
  D)
  \[\frac{x}{\sqrt{1+{{x}^{2}}}}\] done clear
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• question_answer19)
  The value of c + 2 for which the area of the figure bounded by the curve\[y=8{{x}^{2}}-{{x}^{5}}\], the straight lines \[x=1\] and \[x=c\] and x-axis is equal to \[\frac{16}{3},\] is
  A)
  1 done clear
  B)
  3 done clear
  C)
  -1 done clear
  D)
  4 done clear
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• question_answer20)
  What is the area bounded by y = tan x, y = 0 and\[x=\frac{\pi }{4}\]?
  A)
  \[\ell \,n\,\,2\] square units done clear
  B)
  \[\frac{\ell \,n\,\,2}{2}\]square units done clear
  C)
  \[2(\ell n\,2)\] square units done clear
  D)
  None of these done clear
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• question_answer21)
  What is the area of the portion of the curve\[y=sin\text{ }x\], lying between \[x=0,\text{ }y=0\] and\[x=2\pi \]?
  A)
  1 square unit done clear
  B)
  2 square units done clear
  C)
  4 square units done clear
  D)
  8 square units done clear
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• question_answer22)
  What is the area enclosed between the curves\[{{y}^{2}}=12x\] and the lines \[x=0\] and \[y=6\]?
  A)
  2 sq. unit done clear
  B)
  4 sq. unit done clear
  C)
  6 sq. unit done clear
  D)
  8 sq. unit done clear
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• question_answer23)
  What is the area bounded by the curves \[y={{e}^{x}},y={{e}^{-x}}\] and the straight line\[x=1\]?
  A)
  \[\left( e+\frac{1}{e} \right)\] sq. unit done clear
  B)
  \[\left( e-\frac{1}{e} \right)\] sq. unit done clear
  C)
   \[\left( e+\frac{1}{e}-2 \right)\] sq. unit done clear
  D)
  \[\left( e-\frac{1}{e}-2 \right)\] sq. unit done clear
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• question_answer24)
  The area bounded by the curve \[y=x{{(3-x)}^{2}}\], the x-axis and the ordinates of the maximum and minimum points of the curve, is given by
  A)
  1 sq. unit done clear
  B)
  2 sq. unit done clear
  C)
  4 sq. unit done clear
  D)
  None of these done clear
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• question_answer25)
  The area bounded by the x-axis, the curve \[y=f(x)\] and the lines \[x=1,\text{ }x=b,\] is equal to\[\sqrt{{{b}^{2}}+1}-\sqrt{2}\]  for all \[b>1\], then f(x) is
  A)
  \[\sqrt{x-1}\] done clear
  B)
  \[\sqrt{x+1}\] done clear
  C)
  \[\sqrt{{{x}^{2}}+1}\] done clear
  D)
  \[\frac{x}{\sqrt{1+{{x}^{2}}}}\] done clear
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• question_answer26)
  Which of the following is not the area of the region bounded by \[y={{e}^{x}}\] and \[x=0\] and y = e?
  A)
  \[e-1\] done clear
  B)
  \[\int\limits_{1}^{e}{ln(e+1-y)dy}\] done clear
  C)
  \[e-\int\limits_{0}^{1}{{{e}^{x}}dx}\] done clear
  D)
  \[\int\limits_{1}^{e}{\,ln\,\,y\,\,dy}\] done clear
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• question_answer27)
  The area enclosed by the curve \[x=a\text{ }co{{s}^{3}}t,\]\[y=b\text{ }si{{n}^{3}}t\] and the positive directions of x-axis and y-axis is
  A)
  \[\frac{\pi ab}{4}\] done clear
  B)
  \[\frac{\pi ab}{32}\] done clear
  C)
  \[\frac{3\pi ab}{32}\] done clear
  D)
  \[\frac{5\pi ab}{32}\] done clear
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• question_answer28)
  The area of the region enclosed by the curves \[y=x\,\,\log \,\,x\] and \[y=2x-2{{x}^{2}}\] is  
  A)
  \[\frac{5}{12}\] done clear
  B)
  \[\frac{7}{12}\] done clear
  C)
  1 done clear
  D)
  \[\frac{4}{7}\] done clear
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• question_answer29)
  The area bounded by the curves \[y=f(x),\] the x-axis, and the ordinates \[x=1\]and \[x=b\] is \[(b-1)\sin (3b+4)\]. Then \[f(x)\] is
  A)
  \[(x-1)\cos (3x+4)\] done clear
  B)
  \[sin(3x+4)\] done clear
  C)
  \[\sin (3x+4)+3(x-1)\cos (3x+4)\] done clear
  D)
  None of these done clear
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• question_answer30)
  The area of the region formed by\[{{x}^{2}}+{{y}^{2}}-6x-4y+12\le 0,y\le x\] and \[x\le \frac{5}{2}\] is
  A)
  \[\left( \frac{\pi }{6}-\frac{\sqrt{3}+1}{8} \right)sq\,\,unit\] done clear
  B)
  \[\left( \frac{\pi }{6}+\frac{\sqrt{3}-1}{8} \right)sq\,\,unit\] done clear
  C)
  \[\left( \frac{\pi }{6}-\frac{\sqrt{3}-1}{8} \right)sq\,\,unit\] done clear
  D)
  None of theses done clear
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• question_answer31)
  The area enclosed by the curve \[{{x}^{2}}y=36,\] the x-axis and the lines x = 6 and x = 9 is
  A)
  6 done clear
  B)
  1 done clear
  C)
  4 done clear
  D)
  2 done clear
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• question_answer32)
  Area bounded by the curves \[y={{e}^{x}},y={{e}^{-x}}\] and the straight line \[x=1\] is (in sq. units)
  A)
  \[e+\frac{1}{e}\] done clear
  B)
  \[e+\frac{1}{e}+2\] done clear
  C)
  \[e+\frac{1}{e}-2\] done clear
  D)
  \[e-\frac{1}{e}+2\] done clear
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• question_answer33)
  The area of the region bounded by the parabola \[{{(y-2)}^{2}}=x-1\], the tangent of the parabola at the point (2, 3) and the x-axis is:
  A)
  6 done clear
  B)
  9 done clear
  C)
  12 done clear
  D)
  3 done clear
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• question_answer34)
  Area bounded by the curve \[x{{y}^{2}}={{a}^{2}}(a-x)\]and y-axis is
  A)
  \[\pi {{a}^{2}}/2\] sq. units done clear
  B)
  \[\pi {{a}^{2}}\] sq. units done clear
  C)
  \[3\pi {{a}^{2}}\] sq. units done clear
  D)
  None of these done clear
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• question_answer35)
  The figure shows as triangle AOB and the parabola\[y={{x}^{2}}\]. The ratio of the area of the triangle AOB to the area of the region AOB of the parabola \[y={{x}^{2}}\] is equal to
  A)
  \[\frac{3}{5}\] done clear
  B)
  \[\frac{3}{4}\] done clear
  C)
  \[\frac{7}{8}\] done clear
  D)
  \[\frac{5}{6}\] done clear
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• question_answer36)
  Let f(x) be a continuous function such that the area bounded by the curve y = f(x), x-axis and the lines x = 0 and x = a is \[\frac{{{a}^{2}}}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos a\], then \[f\left( \frac{\pi }{2} \right)=\]
  A)
  1 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[\frac{1}{3}\] done clear
  D)
  None of these done clear
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• question_answer37)
  If \[{{c}_{1}}=y=\frac{1}{1+{{x}^{2}}}\] and \[{{c}_{2}}=y=\frac{{{x}^{2}}}{2}\] be two curves lying in XY-plane, then
  A)
  Area bounded by curve \[y=\frac{1}{1+{{x}^{2}}}\] and \[y=0\] is \[\frac{\pi }{2}\] done clear
  B)
  Area bounded by \[{{c}_{1}}\] and \[{{c}_{2}}\] is \[\frac{\pi }{2}-1\] done clear
  C)
  Area bounded by \[{{c}_{1}}\] and \[{{c}_{2}}\] is \[1-\frac{\pi }{2}\] done clear
  D)
  Area bounded by curve \[y=\frac{1}{1+{{x}^{2}}}\] and x-axis is \[\frac{\pi }{2}\] done clear
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• question_answer38)
  The area bounded by the curves y = ln x, y = ln \[\left| x \right|,y=\left| ln\text{ }x \right|\] and \[,y=\left| ln\text{ }\left| x \right| \right|\] is
  A)
  4 sq. units done clear
  B)
  6 sq. units done clear
  C)
  10 sq. units done clear
  D)
  None of these done clear
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• question_answer39)
  If the ordinate \[x=a\] divides the area bounded by x-axis, part of the curve \[y=1+\frac{8}{{{x}^{2}}}\] and the ordinates \[x=2,\text{ }x=4\] into two equal parts, then a is equal to
  A)
  \[\sqrt{2}\] done clear
  B)
  \[2\sqrt{2}\] done clear
  C)
  \[3\sqrt{2}\] done clear
  D)
  None of these done clear
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• question_answer40)
  If the area enclosed by \[{{y}^{2}}=4ax\] and line \[y=ax\]is 1/3 sq. units , then the area enclosed by \[y=4x\]with same parabola is
  A)
  8 sq. units done clear
  B)
  4 sq. units done clear
  C)
  4/3 sq. units done clear
  D)
  8/3 sq. units done clear
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• question_answer41)
  The area bounded by the curve \[y=si{{n}^{-1}}x\] and the line \[x=0,\left| y \right|=\frac{\pi }{2}\] is
  A)
  1 done clear
  B)
  2 done clear
  C)
  \[\pi \] done clear
  D)
  \[2\pi \] done clear
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• question_answer42)
  The area of the region\[R=\{(x,y):\left| x \right|\le \left| y \right|\] and \[{{x}^{2}}+{{y}^{2}}\le 1\}\] is
  A)
  \[\frac{3\pi }{8}\] sq. unit done clear
  B)
  \[\frac{5\pi }{8}\] sq. unit done clear
  C)
  \[\frac{\pi }{2}\] sq. unit done clear
  D)
  \[\frac{\pi }{8}\] sq. unit done clear
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• question_answer43)
  The area bounded by \[f(x)={{x}^{2}},0\le x\le 1,\] \[g(x)=-x+2,1\le x\le 2\] and \[x-axis\] is
  A)
  \[\frac{3}{2}\] done clear
  B)
  \[\frac{4}{3}\] done clear
  C)
  \[\frac{8}{3}\] done clear
  D)
  None of these done clear
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• question_answer44)
  The area bounded by the curves \[{{x}^{2}}+{{y}^{2}}=25,\]\[4y=\left| 4-{{x}^{2}} \right|\] and \[x=0\], above x-axis is
  A)
  \[2+\frac{25}{2}{{\sin }^{-1}}\frac{4}{5}\] done clear
  B)
  \[2+\frac{25}{4}{{\sin }^{-1}}\frac{4}{5}\] done clear
  C)
  \[2+\frac{25}{2}{{\sin }^{-1}}\frac{1}{5}\] done clear
  D)
  None of these done clear
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• question_answer45)
  What is the area bounded by the curve \[y=4x-{{x}^{2}}-3\] and the x-axis?
  A)
  2/3 sq. unit done clear
  B)
  4/3 sq. unit done clear
  C)
  5/3 sq. unit done clear
  D)
  4/5 sq. unit done clear
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• question_answer46)
  What is the area enclosed by the equation\[{{x}^{2}}+{{y}^{2}}=2\]?
  A)
  \[4\pi \] square units done clear
  B)
  \[2\pi \] square units done clear
  C)
  \[4{{\pi }^{2}}\] square units done clear
  D)
  4 square units done clear
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• question_answer47)
  The line y = mx bisects the area enclosed by lines \[x=0,\text{ }y=0\] and \[x=3/2\] and the curve\[y=1+4x-{{x}^{2}}\]. Then the value of m is
  A)
  \[\frac{13}{6}\] done clear
  B)
  \[\frac{13}{2}\] done clear
  C)
  \[\frac{13}{5}\] done clear
  D)
  \[\frac{13}{7}\] done clear
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• question_answer48)
  The area enclosed between the curves \[y=a{{x}^{2}}\] and \[x=a{{y}^{2}}(a>0)\] is 1 sq. unit, then the value of a is
  A)
  \[\frac{1}{\sqrt{3}}\] done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  1 done clear
  D)
  \[\frac{1}{3}\]    done clear
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• question_answer49)
  If \[y=f(x)\] makes +ve intercept of 2 and 0 unit on x and y axes and encloses an area of 3/4 square unit with the axes then \[\int\limits_{0}^{2}{xf'(x)dx}\] is
  A)
  3/2 done clear
  B)
  1 done clear
  C)
  5/4 done clear
  D)
  -3/4 done clear
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• question_answer50)
  The slope of the tangent to a curve \[y=f(x)\] at \[(x,f(x))\] is\[2x+1\]. If the curve passes through the point (1, 2), then the area of the region bounded by the curve, the x-axis and the line \[x=1\] is
  A)
  \[\frac{5}{6}\] sq. unit done clear
  B)
  \[\frac{6}{5}\] sq. unit done clear
  C)
  \[\frac{1}{6}\] sq. unit done clear
  D)
  6 sq. unit done clear
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