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Applications of Integration Three Marks Questions

12th Standard EM

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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.