Integration(Senior Class Math Study by Topics Test ) : GATE Academy

Welcome to your Integration(Senior Class Math Study by Topics)

NAME

PHONE NUMBER

1. Given that dv/dt = -64t and v = 10 when t = 0, find v when t = 2.

v = -64

v = 118

v = 64

v = -118

2. Find the area bounded by the curve y = x(2 - x), the x-axis, x=0 and x= 2.

4/3

3. Evaluate the definite integral:

102

4. Find
∫ 1/ x²+3x-10 dx

1/7 In (x-2 / x-5) + C

7In x + C

In (x-2 / x-5) + C

5. The gradient of a curve is given by dy/dx = 10x - 3x². If the curve passes through the origin, find the equation of the curve.

y = 5x² - x³

y = x³ - 5x²

y = 5x² + x³

y = x³ + 5x²

6. Given that y and x are two variables such that dy/dx = 2x - 6, and y = -1 when x = 3, express y in terms of x.

y = x² + 6x + 8

y = x² + 6

y = x² - 6x + 8

y = x - 3

7. The acceleration of a moving particle at the end of t seconds is given by (5 - t) m/s². If the particle starts from rest, find its velocity at the end of 4 s?

12 m/s

4 m/s

24 m/s

0.5 m/s

8. Find the area bounded by the curve y = x³ -3x² -x + 3 and the x-axis

2.1

3.3

9. Evaluate the definite integral:

-4

10. The finite are enclosed by the curve y² = 4x and the line x = 4 is rotated through 360° about the x-axis.
Find the volume of the solid generated in cubic units.

27π

32π

11. Given that x and t are two variables such that dx/dt = 3t² - 4t, and x = 1 when t = 4, express x in terms of t.

x = 3t³ + 2t² - 31

x = t³ - 2t - 31

x = 3t³ - 2t² - 31

x = t³ - 2t² - 31

12. Evaluate:

12/7

12/5

1/2

14/3

13. Evaluate:

correct to 3 decimal places

2.368

3.797

2.868

2.386

14. A line with equation y = 4x + 2 intersects a curve with the equation y = x² + 2 at the points M and N. Determine the coordinates of M and N

M(0, 4), N(4, 18)

M(2, 5), N(3, 18)

M(2, 3), N(2, 5)

M(1, 3), N(3, 12)

15. Calculate the area under the line
y = x + 2 between the limits x = 0 and x = 10

16. The gradient of a curve that passes through the point (-2, 6) is given by x. Determine the equation of the curve.

y = -1/2 x² - 4

y = x² + 4

y = x² - 4

y = 1/2 x² + 4

17. Evaluate the definite integral:

45/2

33/4

32/5

18. Evaluate the integrals:

-1

19. Find
∫ x Inx dx

xIn x + C

xe +C

x² Inx - x²/4 + C

Inx + C

20. Find

∫(x + 1)(x² - 2)dx

2x - x + C

x² - 2x + C

x³ + 2x + C

x4/4 + x³/3 - x² - 2x + C

21. Find
∫ sin x cos 3x dx

1/8 cos4x + 1/4 sin2x + C

-1/8 cos4x + 1/4 sin2x + C

1/4 cos2x + C

-1/8 cos4x + 1/4 cos2x + C

22. The gradient of a curve which passes through the point (1, -5) is given by 4x. Determine the equation of the curve.

y = x² - 14

y = -2x² - 7

y = 2x² - 7

y = -2x² + 7

23. Evaluate:

8In 5

4/5In 3

10/3 - 5In3

6- 2In5

24. Evaluate
∫ x / x+3 dx

x-3 + C

In(x + 3) + C

x - 3 In(x+3) + C

25. A line with equation y = 4x + 2 intersects a curve with the equation y = x² + 2 at the points M and N. Evaluate the area enclosed between the curve and the line

21/4

32/3

11/2

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