This document discusses various applications of integration, including: 1) Velocity and distance problems involving motion along a line 2) Calculating the area of different regions bounded by curves 3) Finding volumes of solids obtained by rotating areas about axes 4) Determining arc lengths and lengths of curves 5) Calculating velocity and acceleration vectors of objects moving along planar curves 6) Applying integrals to problems involving work, average values, forces, and other rates of change.

1. Application of Integration

2. Application of Integration
posted on: 18 Jan, 2012 | updated on: 05 Jun, 2012
Topics Covered in Application of Integration
Velocity and Distance Problems Involving Motion Along a Line
Area of Region | Area of a Region
Area Between Curves | Areas Between Curves
Volumes of Solid of Revolutions
Area of Region Bounded by Parametrically Defined or Polar Curves
Arc Length | Calculus Arc Length
Velocity and Acceleration Vectors of Planar curves
Other Applications involving the use of Integral of Rates

3. Velocity and Distance Problems Involving Motion Along a Line
posted on: 13 Mar, 2012 | updated on: 05 Jun, 2012
Area of region
posted on: 06 Feb, 2012 | updated on: 24 May, 2012

4. Calculate the area of the region given by the following lines: a + b
= 2, a – b = - 1, a + 2 y = 2 ?

5. Find the area of the Region bounded by the following set of
equations: m = n 3 – 6 n 2 – 1 6 n and m = 8 n + 2 n 2 – n 3?

6. Find the area of the region enclosed between the following
curves, a = b 2 – 2 b + 2 and a = - b 2 + 6 ?

7. Find the area enclosed by the ellipse: X / m 2+ y / n 2 = 1?

8. Find the area of the region bounded by the curves y=x and x2 +y
2
= 32, common with first quadrant?

9. Find the area between the given curves, x 2 + y 2 = 4 and ( x – 2 ) 2
+ y 2 = 4?

10. Find the area of the curve x = 2 - 2 sin ??

11. Find the area between the curves n = 1 and n = 2 sin ??

12. Find the area bounded by the following equations: L (x) = 4 x – x2
and n (x) = 5 - 2x?

13. Calculate the area underneath the curve y = x2 + 2 from x = 1 to x
= 2?

14. FAQ of Area of Region | Area of a Region
Find the Area of the Region?
Arc Length of a Circle
Area between curves
posted on: 06 Feb, 2012 | updated on: 24 May, 2012

15. Volumes of Solid of Revolutions
posted on: 06 Feb, 2012 | updated on: 24 May, 2012

16. Calculate the volume if area bounded by curve is y = x3 + 1, limits
are x = 0 and x = 3 and ‘x’ axis are rotated around x – axis?
Area of Region Bounded by Parametrically Defined or Polar Curves
posted on: 06 Feb, 2012 | updated on: 24 May, 2012

17. Arc Length
posted on: 06 Feb, 2012 | updated on: 24 May, 2012

18. Determine the arc length of y = log (cosec x) where x lies
between 0 to ?/4?
Calculate the arc length of the function f(x) = (x 5) / 2 over the
interval [0,1]?

19. Calculate the arc length of the circle whose radius is given as 5m
and central angle is 300?

20. Calculate the arc length of y = log (sec t) between 0 ? t ? ?/2?

21. Determine the length of the function x = (2 / 5) * (y – 1) * (5 /
2) where y lies between 1 ? y ? 4?

22. Calculate the length of the function x = (1 / 2) y 3 for the values 0
? x ? (1 / 2)?

23. Calculate the length of arc on the given curve y = (x)3, from point
(-1 , -1) to (2 , 8)?

24. FAQ of Arc Length | Calculus Arc Length
Arc Length in Polar Coordinates
Calculate Arc Length?
Calculus Arc Length Formula
Velocity and Acceleration Vectors of Planar curves
posted on: 06 Feb, 2012 | updated on: 24 May, 2012

25. If a particle is moving along a plane curve 'C' then calculate the
velocity vector and acceleration vector. The plane curve 'C' is
described by r(t) = 2 sin t/2 i + 2 cos t/2 j?

26. If a plane curve 'C' is represented by r (t) = (t2 – 4)i + tj then find
the velocity and acceleration vectors when t = 0 and t = 2?

27. Find the velocity and acceleration vectors if an object is moving
along a curve 'C' represented by r (t) = ti + t3j + 3tk, t ? 0?

28. An object moves in xy plane at any time 't', the position of object
is given by x(t) = t3 + 4t2, y(t) = t4 – t3. Calculate the velocity vector
when t = 1 and acceleration vector when t = 2?

29. A body is moving in a plane and its position at any time t ? 0 is
(sin t, t2/2). Calculate the velocity vector and acceleration vector
of the moving body?

30. If an object moves in a plane and has position vector r (t) = [sin
(3t), cos (5t)]. Calculate the velocity and acceleration vectors?

31. If a particle moves along a plane curve having position vector r
(t) = [4 sin t, 9 cos t]. Calculate the velocity vector and
acceleration vector?

32. An object moves in x-y plane at any time 't', the position of object
is given by x(t) = t4 + 3t, y(t) = t3 – t2. Calculate the velocity vector
when t = 2 and acceleration vector when t = 3?

33. A particle moves in x-y plane at any time 't', the position vector of
particle is given by x(t) = t3 +1, y(t)= t2. Calculate the velocity
vector and acceleration vector?

34. When a body is moving in x-y plane at any time ‘t’, the position
vector of body is given by x(t) = t5, y(t) = t3. Calculate the velocity
vector and acceleration vector?

35. Other Applications involving the use of Integral of Rates
posted on: 06 Feb, 2012 | updated on: 24 May, 2012

36. Calculate the amount of work done on a spring, when spring is
compressed from its natural length of 1 unit to a length of 0.75
units, if the spring constant is equals to k = 16?
A spring is compressed by a force of 1200 N from its natural
length of 18 units to a length of 16 units. Calculate the amount of
work done in compressing it from 16 units to 14 units?

37. The temperature recorded during the day follows the curve T =
0.001 t4 – 0.280 t2
+ 25, where ' t ' is the number of hours from
noon. (- 1 ? t ? 2). Calculate the average temperature of during
the day?

38. A plate with right triangular base of 2.0 units and height 1.0 units
is vertically submerged, with the top vertex 3.0 units below the
surface. Calculate the force on one side of place?

39. The movement of proton in an electric field with acceleration a = -
20 (1 + 2t)- 2, where time 't' is in seconds. Calculate the velocity as
a function of time if v = 30 units / s when t = 0?

40. A flare is launched vertically upwards from surface at 15 unit / s.
Calculate the height of flare after 2.5 s?

41. The electric current as a function of time is given by i = 0.3 – 0.2
t, in a computer circuit. Calculate the amount of charge passes
through a point in circuit in 0.50 s?

42. A 8.50 nf capacitor has a voltage of zero in an FM receiver.
Calculate the voltage after 2.00 ?s if a current i = 0.042t (in mA)
charges the capacitor?

43. The initial velocity of moving car is 5 mph and its acceleration is
a ( t ) = 2.4 t mph for 8 seconds. Calculate the velocity of car
when 8 seconds are up?

44. The initial velocity of a moving car is 5 mph with rate of
acceleration a (t) = 2.4 t mph for 8 seconds. Calculate the
distance covered by car during those 8 seconds?

45. The amount of force required to stretch a spring by 2 units
beyond its natural length. Calculate the amount of work done to
stretch the spring 4 units from its natural length?

46. Between the year 1970 to 1980, the rate of consumption of potato
in a country was R (t) = 2.2 + 1.1 t millions bushels per year, while
't' are the years from beginning of 1970. Calculate the
consumption of from start of 1972 to the end of 1973?

47. If the acceleration of a body is given by a = t2 + 1, between time
interval t = 2 to t = 3, then calculate the velocity of the body in the
given interval?

49. Further Read
Application of Integration Examples
Application of Integration FAQs
Application of Integration Worksheets
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