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#### Applications of Integration Model Question Paper

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 40
5 x 1 = 5
1. The value of $\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx }$ is

(a)

$\pi$

(b)

$2\pi$

(c)

$3\pi$

(d)

$4\pi$

2. The value of $\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \\$ is

(a)

$\frac{7}{27}$

(b)

$\frac{5}{27}$

(c)

$\frac{4}{27}$

(d)

$\frac{2}{27}$

3. The value of $\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ \sqrt { 4-9{ x }^{ 2 } } } }$ is

(a)

$\frac{\pi}{6}$

(b)

$\frac{\pi}{2}$

(c)

$\frac{\pi}{4}$

(d)

${\pi}$

4. If $\int _{ 0 }^{ 2a }{ f(x) } dx=2\int _{ 0 }^{ a }{ f(x) }$ then

(a)

f(2a -x) = - f(x)

(b)

f(2a - x) = f(x)

(c)

f(x) is odd

(d)

f(x) is even

5. The ratio of the volumes generated by revolving the ellipse $\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 }$ = 1 about major and minor axes is

(a)

4:9

(b)

9:4

(c)

2:3

(d)

3:2

6. 5 x 2 = 10
7. Evaluate the following definite integrals:
$\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } }$

8. Evaluate the following integrals using properties of integration:
$\int _{ 0 }^{ 2\pi }{ { sin }^{ 4 }{ x\quad cos }^{ 3 }dx }$

9. Evaluate the following:
$\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ e }^{ -2x }dx }$

10. Evaluate $\int _{ 0 }^{ 1 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx }$

11. Find the area of the region enclosed by the curve y = $\sqrt x$ + 1, the axis of x and the lines x=0, x=4.

12. 5 x 3 = 15
13. Evaluate $\int _{ 1 }^{ 2 }{ \frac { x }{ (x+1)(x+2) } dx }$

14. Evaluate:  $\int ^{\frac{\pi}{2}}_{\frac{\pi}{2}}$x cox x dx.

15. 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.

16. Evaluate $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sin \ x }{ 9+{ cos }^{ 2 } } dx }$

17. Evaluate $\int _{ 0 }^{ 1 }{ \sqrt { 9-4{ x }^{ 2 } } dx }$

18. 2 x 5 = 10
19. Evaluate $\int ^\frac {\pi}{2}_{0}$( sin2 x + cos4 x ) dx

20. Find the area of the region bounded by x−axis, the curve y = cos x, the lines x = 0 and x =$\pi$.