Integral Calculus

Question 1

Set 1 Problem 67: Find the length of the curve y=lnx from (1,0) to (4,ln4).

Answer 1

3.34 (p.17)

Question 2

Set 2 Problem 14: Find the area which is inside the circle r=3sin⁡θ and outside the cardioid r=1+sin⁡θ.

Answer 2

π sq. units (p. 24)

Question 3

Set 2 Problem 26: Find the area bounded by the curves x²y=4 and y=7-3x.

Answer 3

1/2 (p. 27)

Question 4

Set 2 Problem 66: The area bounded by y^3=x^2, y=4 and the y-axis is revolved about the y-axis. Find the centroid of the solid generated.

Answer 4

16/5 (p. 33)

Question 5

Set 2 Problem 68: Find the area bounded by the curves y=(x-1)^3 and y=x^2-x-1.

Answer 5

4/3 sq. units (p. 34)

Question 6

Set 2 Problem 69: Find the volume of the solid generated by revolving about the x-axis the smaller area bounded by the circle x^2+y^2=2 and the semi-cubical parabola y^3=x^2.

Answer 6

52pi/21 cubic units (p. 34)

Question 7

Set 3 Problem 31: Find the area of the surface generated by revolving about the y-axis the arc of y=x^2 from x=0 to x=6/5.

Answer 7

(1036/375)pi (p. 46)

Question 8

Set 3 Problem 34: A spring whose normal length is 10 inches has a modulus of 12 pounds per inch. How much work in in-lb is done in stretching this spring from a length of 12 inches to a length of 15 inches?

Answer 8

126 in-lb (p. 47)

Question 9

Set 3 Problem 52: A particle moves in the xy-plane so that its acceleration vector for time, t>0 is given as (12t^2,10/t), where time is in seconds and acceleration in m/s^2. If the velocity vector at t=1 is (4,0), how fast is the particle moving when t=3 seconds?

Answer 9

108.56 m/s (p. 51)

Question 10

Set 3 Problem 65: The arc if y=x^3 from x=1 to y=2 is revolved about the y-axis. Find the area of the surface generated.

Answer 10

71.41 sq. units (p. 53)

Question 11

Set 4 Problem 15: Compute the area bounded by the curves y^2=4x and x^2=4y.

Answer 11

5.33 sq. units (p.62)

Question 12

Set 4 Problem 69: Find the area bounded by the parabolas x=y²-4y and x=2y-y².

Answer 12

9 sq. units (p.70)

Question 13

Set 4 Problem 70: Find the area bounded by the parabolas x=y²-4y and x=2y-y².

Answer 13

pi/2 cubic units (p.71)

Question 14

Set 5 Problem 3: Find the length of one arc of the curve whose equation is represented in parametric form: x =8(θ-sin⁡θ), y=8(1-cos⁡θ).

Answer 14

64 (p.77)

Question 15

Set 5 Problem 11: Find the volume obtained if the region bounded by y=4x-x^2 and y=x is rotated about the x=3.

Answer 15

27pi/2 sq. units (p.77)

Question 16

Set 5 Problem 64: Evaluate the integral of arctanxdx from x=0 to x=1.

Answer 16

0.439 (p.87)

Question 17

Set 5 Problem 68: Find the particular solution of the differential: dy/dx-3y/x=(x cubed); y(1)=4.

Answer 17

y=(x to the 4th power)+3(x cubed) (p.88)

Question 18

Set 5 Problem 69: Find the area inside the curve r^2=2cos2θ and outside the curve r=1.

Answer 18

sqrt(3)-pi/3 (p.88)

Question 19

Set 6 Problem 35: Find the area enclosed by the curve r=2cos3θ.

Answer 19

π sq. units (p.99)

Question 20

Set 6 Problem 72: A tank contains 100 liters of brine with 25 kg of dissolved salt. Brine containing 0.5 kg per liter enters the tank at the rate of 2 liters per minute, and the well-stirred mixture runs out at the same rate. How many kilograms of salt are in the tank after 30 minutes?

Answer 20

36.28 kg (p.105)

Question 21

Set 7 Problem 1: Find the length of the curve y=sinx from x=0 to x=π.

Answer 21

3.82 (p.111)

Question 22

Set 7 Problem 41: Evaluate: Equation 7.1.

Answer 22

1.414 (p.117)

Question 23

Set 7 Problem 63: Evaluate: Equation 7.3?

Answer 23

7.34 (p.121)

Question 24

Set 7 Problem 65: The half-life of a certain radioactive substance is 6.4 years. What percent of the original amount of the substance will be left after 10 years?

Answer 24

33.86% (p.121)

Question 25

Set 8 Problem 10: A bottle of soda pop at room temperature 22degC is placed in a refrigerator where the temperature is 7degC. After half an hour the soda pop has cooled to 16degC. How long does it take for the soda pop to cool 10degC.

Answer 25

1.6 hrs (p.129)

Question 26

Set 8 Problem 16: A solid is formed by revolving about the y-axis the area bounded by y^3=x^2, x=0, and y=4. Find the volume of the solid generated.

Answer 26

64pi (p.130)

Question 27

Set 8 Problem 20: Find the equation of the curve which passes through (2,-1) satisfying the condition that at any of its point (x,y) the slope is x²/(y-1).

Answer 27

2x^3-3y²+6y=7 (p.131)

Question 28

Set 8 Problem 21: A right circular cylindrical tank of radius 4 ft and height of 12 ft is half full of oil weighing 60 pounds per cubic foot. Find the work done in ft-tons in pumping the oil to a height 6 ft above the top of the tank.

Answer 28

136 ft-tons (p.131)

Question 29

A plane region is bounded by the parabola x^2=16y, the line x=12, and the x-axis. Set 8 Problem 36: Compute the area of the plane region.

Answer 29

36 (p.134)

Question 30

A plane region is bounded by the parabola x^2=16y, the line x=12, and the x-axis. Set 8 Problem 37: Find the distance of the centroid of the plane region from the y-axis.

Answer 30

9 (p.134)

Question 31

A plane region is bounded by the parabola x^2=16y, the line x=12, and the x-axis. Set 8 Problem 38: Compute the moment of inertia of the plane region with respect to the x-axis.

Answer 31

416.57 (p.134)

Question 32

Set 8 Problem 51: Evaluate: Equation 8.1.

Answer 32

112.5 (p.136)

Question 33

In the chemical processing of a certain mineral, the rate of change of the amount of mineral present varies as the amount of the mineral remaining. After 8 hours, 100 pounds of mineral have been reduced to 70 pounds. Set 8 Problem 68: Find the rate at which the amount of mineral is decreasing.

Answer 33

-0.044584 lb/hr (p.139)

Question 34

In the chemical processing of a certain mineral, the rate of change of the amount of mineral present varies as the amount of the mineral remaining. After 8 hours, 100 pounds of mineral have been reduced to 70 pounds. Set 8 Problem 69: What amount of the mineral remains after 24 hours?

Answer 34

34.30 lb (p.139)

Question 35

In the chemical processing of a certain mineral, the rate of change of the amount of mineral present varies as the amount of the mineral remaining. After 8 hours, 100 pounds of mineral have been reduced to 70 pounds. Set 8 Problem 70: After how many hours when there are only 50 pounds of minerals remaining?

Answer 35

15.55 hrs (p.139)

Question 36

Set 8 Problem 72: After touching the ground, the speed (in feet per second) of an airplane is given by v=180-18t. Find how far the airplane will move in landing.

Answer 36

900 ft (p.139)

Question 37

Set 8 Problem 28: Which of the following is the orthogonal trajectories of the family of parabola y²=4ax?

Answer 37

2x^2+y²=C (p.149)

Question 38

Set 9 Problem 43: Evaluate the integral of dx/(2x+3) from x=-1 to x=0.

Answer 38

ln(sqrt of 3) (p.152)

Question 39

Set 9 Problem 49: Find the moment of inertia with respect to the y-axis of the area bounded by x²=4y, y=4 and x=0.

Answer 39

34.13 (p.152)

Question 40

Set 10 Problem 6: Find the area of the region bounded by x=y^3 and the x-axis from x=0 to x=8.

Answer 40

12 sq. units (p.162)

Question 41

Set 10 Problem 12: Find the area bounded by the curves x^2=2y and y=x^2-2x.

Answer 41

16/3 sq. units (p.163)

Question 42

Set 10 Problem 13: The area between the x-axis and x^2=9y from x=2 to x=6 is revolved about the y-axis. Find the volume of the soil generated.

Answer 42

1295pi/18 cubic units (p.164)

Question 43

Set 10 Problem 19: Evaluate the integral of cos³xdx from x=0 to x=π/4.

Answer 43

-(3√2)/8 (p.165)