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Applications of Integration Three Marks Questions

12th Standard EM

    Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 45
    15 x 3 = 45
  1. Estimate the value of \(\int _{ 0 }^{ 0.5 }{ { x }^{ 2 } } dx\)using the Riemann sums corresponding to 5 subintervals of equal width and applying (i) left-end rule (ii) right-end rule (iii) the mid-point rule.

  2. Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)

  3. Evaluate :\(\int _{ 0 }^{ 1 }{ \frac { 2x+7 }{ { 5x }^{ 2 }+9 } } dx\)

  4. Evaluate :\(\int _{ 0 }^{ 9 }{ \frac { 1 }{ x+\sqrt { x } } dx } \)

  5. Show that \(\int ^\frac{\pi}{2}_0\) \(\frac {dx}{4+5 sin x}\) = \(\frac {1}{3}\) log2

  6. Prove that \(\int^{\frac{\pi}{4}}_{0} \frac {sin 2x dx}{ sin ^4x +cos ^4 x}\) = \(\frac{\pi}{4}\) 

  7. Evaluate:  \(\int ^{\frac{\pi}{2}}_{\frac{\pi}{2}}\)x cox x dx.

  8. Evaluate: \(\int ^{log 2}_{-log 2} e ^{-|x|}\) dx.

  9. Show that \(\int ^{1}_{0} (tan ^{-1} x + tan ^{-1}(1-x))\) dx = \(\frac {\pi}{2}\) - loge

  10. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 4{ sin }^{ 2 }x+5{ cos }^{ 2 }x } } \)

  11. Prove that \(\int _{ 0 }^{ \infty }{ { e }^{ -x }{ x }^{ n }dx=n! } \) where n is a positive integer.

  12. Evaluate \(\int _{ 0 }^{ \infty }{ \frac { { x }^{ n } }{ { x }^{ x } } } dx\), where n is positive integer \(\ge\)2

  13. Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded between the parabola x = y2 +1, the y-axis, and the lines y =1 and y = −1.

  14. Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded between the curve y=\(\frac{3}{4} \sqrt {x^2 -16}, x\ge4\)the y-axis, and the lines y =1 and y = 6.

  15. Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded by the curves y = log x, y = 0, x = 0 and y = 2.

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