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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 : 45
5 x 1 = 5
1. The value of $\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3xdx }$

(a)

$\frac{2}{3}$

(b)

$\frac{2}{9}$

(c)

$\frac{1}{9}$

(d)

$\frac{1}{3}$

2. If f(x)$f(x)=\int _{ 1 }^{ x }{ \frac { { e }^{ { sin }^{ u } } }{ u } } du,x>1\quad and\quad \int _{ 1 }^{ 3 }{ \frac { { e }^{ { sinx }^{ 2 } } }{ x } } dx=\frac { 1 }{ 2 } [f(a)-f(1)]$, then one of the possible value of a is

(a)

3

(b)

6

(c)

9

(d)

5

3. The value of $\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }xcosxdx }$ is

(a)

$\frac{3}{2}$

(b)

$\frac{1}{2}$

(c)

0

(d)

$\frac{2}{3}$

4. The area enclosed by the curve y = $\frac { { x }^{ 2 } }{ 2 }$ , the x - axis and the lines x = 1, x = 3 is

(a)

4

(b)

8$\frac23$

(c)

13

(d)

4$\frac{1}{3}$

5. The area enclosed by the curve y2 = 4x, the x-axis and its latus rectum is ............ sq.units.

(a)

$\frac23$

(b)

$\frac43$

(c)

$\frac83$

(d)

$\frac{16}{3}$

6. 2 x 2 = 4
7. The area of the region bounded by the graph of y = sin x and y = cos x between x = 0 and x = $\frac { \pi }{ 4 }$
(1) $\int _{ 0 }^{ \frac { \pi }{ 4 } }{ (cos \ x-sin \ x) } dx$
(2) ${ \left[ sinx+cosx \right] }_{ 0 }^{ \frac { \pi }{ 4 } }$
(3) $\int _{ 0 }^{ \frac { \pi }{ 4 } }{ (sinx-cosx) } dx$
(4) $\sqrt { 2 } -1$

8. $\int _{ a }^{ b }{ f(x) } dx=$
(1) $\int _{ a }^{ b }{ f(y) } dy$
(2) $-\int _{ a }^{ b }{ f(x) } dx$
(3) $\int _{ a }^{ b }{ f(a+b-x) } dx$
(4) $\int _{ 0 }^{ a }{ f(a-x) } dx$​​

9. 3 x 2 = 6
10. Evaluate:  $\int ^{\frac{\pi}{2}}_{\frac{\pi}{2}}$x cox x dx.

11. Evaluate the following definite integrals:
$\int _{ 3 }^{ 4 }{ \frac { dx }{ { x }^{ 2 }-4 } }$

12. 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.

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

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

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

17. 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.

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

19. 3 x 5 = 15
20. Show that $\int ^\frac{\pi}{2}_0$ $\frac {dx}{4+5 sin x}$ = $\frac {1}{3}$ log2

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

22. Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = $\frac { \pi }{ 3 }$ and x axis are as 2:3