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.
Introduction to differential calculus. This slide deals with the concept of differential calculus. An explanation is given that how derivative of a function can be calculated graphically and algebraically
Introduction to differential calculus. This slide deals with the concept of differential calculus. An explanation is given that how derivative of a function can be calculated graphically and algebraically
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
MA 243 Calculus III Fall 2015 Dr. E. JacobsAssignmentsTh.docxinfantsuk
MA 243 Calculus III Fall 2015 Dr. E. Jacobs
Assignments
These assignments are keyed to Edition 7E of James Stewart’s “Calculus” (Early Transcendentals)
Assignment 1. Spheres and Other Surfaces
Read 12.1 - 12.2 and 12.6
You should be able to do the following problems:
Section 12.1/Problems 11 - 18, 20 - 22 Section 12.6/Problems 1 - 48
Hand in the following problems:
1. The following equation describes a sphere. Find the radius and the coordinates of the center.
x2 + y2 + z2 = 2(x + y + z) + 1
2. A particular sphere with center (−3, 2, 2) is tangent to both the xy-plane and the xz-plane.
It intersects the xy-plane at the point (−3, 2, 0). Find the equation of this sphere.
3. Suppose (0, 0, 0) and (0, 0, −4) are the endpoints of the diameter of a sphere. Find the
equation of this sphere.
4. Find the equation of the sphere centered around (0, 0, 4) if the sphere passes through the
origin.
5. Describe the graph of the given equation in geometric terms, using plain, clear language:
z =
√
1 − x2 − y2
Sketch each of the following surfaces
6. z = 2 − 2
√
x2 + y2
7. z = 1 − y2
8. z = 4 − x − y
9. z = 4 − x2 − y2
10. x2 + z2 = 16
Assignment 2. Dot and Cross Products
Read 12.3 and 12.4
You should be able to do the following problems:
Section 12.3/Problems 1 - 28 Section 12.4/Problems 1 - 32
Hand in the following problems:
1. Let u⃗ =
⟨
0, 1
2
,
√
3
2
⟩
and v⃗ =
⟨√
2,
√
3
2
, 1
2
⟩
a) Find the dot product b) Find the cross product
2. Let u⃗ = j⃗ + k⃗ and v⃗ = i⃗ +
√
2 j⃗.
a) Calculate the length of the projection of v⃗ in the u⃗ direction.
b) Calculate the cosine of the angle between u⃗ and v⃗
3. Consider the parallelogram with the following vertices:
(0, 0, 0) (0, 1, 1) (1, 0, 2) (1, 1, 3)
a) Find a vector perpendicular to this parallelogram.
b) Use vector methods to find the area of this parallelogram.
4. Use the dot product to find the cosine of the angle between the diagonal of a cube and one of
its edges.
5. Let L be the line that passes through the points (0, −
√
3 , −1) and (0,
√
3 , 1). Let θ be the
angle between L and the vector u⃗ = 1√
2
⟨0, 1, 1⟩. Calculate θ (to the nearest degree).
Assignment 3. Lines and Planes
Read 12.5
You should be able to do the following problems:
Section 12.5/Problems 1 - 58
Hand in the following problems:
1a. Find the equation of the line that passes through (0, 0, 1) and (1, 0, 2).
b. Find the equation of the plane that passes through (1, 0, 0) and is perpendicular to the line in
part (a).
2. The following equation describes a straight line:
r⃗(t) = ⟨1, 1, 0⟩ + t⟨0, 2, 1⟩
a. Find the angle between the given line and the vector u⃗ = ⟨1, −1, 2⟩.
b. Find the equation of the plane that passes through the point (0, 0, 4) and is perpendicular to
the given line.
3. The following two lines intersect at the point (1, 4, 4)
r⃗ = ⟨1, 4, 4⟩ + t⟨0, 1, 0⟩ r⃗ = ⟨1, 4, 4⟩ + t⟨3, 5, 4⟩
a. Find the angle between the two lines.
b. Find the equation of the plane that contains every point o ...
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
MA 243 Calculus III Fall 2015 Dr. E. JacobsAssignmentsTh.docxinfantsuk
MA 243 Calculus III Fall 2015 Dr. E. Jacobs
Assignments
These assignments are keyed to Edition 7E of James Stewart’s “Calculus” (Early Transcendentals)
Assignment 1. Spheres and Other Surfaces
Read 12.1 - 12.2 and 12.6
You should be able to do the following problems:
Section 12.1/Problems 11 - 18, 20 - 22 Section 12.6/Problems 1 - 48
Hand in the following problems:
1. The following equation describes a sphere. Find the radius and the coordinates of the center.
x2 + y2 + z2 = 2(x + y + z) + 1
2. A particular sphere with center (−3, 2, 2) is tangent to both the xy-plane and the xz-plane.
It intersects the xy-plane at the point (−3, 2, 0). Find the equation of this sphere.
3. Suppose (0, 0, 0) and (0, 0, −4) are the endpoints of the diameter of a sphere. Find the
equation of this sphere.
4. Find the equation of the sphere centered around (0, 0, 4) if the sphere passes through the
origin.
5. Describe the graph of the given equation in geometric terms, using plain, clear language:
z =
√
1 − x2 − y2
Sketch each of the following surfaces
6. z = 2 − 2
√
x2 + y2
7. z = 1 − y2
8. z = 4 − x − y
9. z = 4 − x2 − y2
10. x2 + z2 = 16
Assignment 2. Dot and Cross Products
Read 12.3 and 12.4
You should be able to do the following problems:
Section 12.3/Problems 1 - 28 Section 12.4/Problems 1 - 32
Hand in the following problems:
1. Let u⃗ =
⟨
0, 1
2
,
√
3
2
⟩
and v⃗ =
⟨√
2,
√
3
2
, 1
2
⟩
a) Find the dot product b) Find the cross product
2. Let u⃗ = j⃗ + k⃗ and v⃗ = i⃗ +
√
2 j⃗.
a) Calculate the length of the projection of v⃗ in the u⃗ direction.
b) Calculate the cosine of the angle between u⃗ and v⃗
3. Consider the parallelogram with the following vertices:
(0, 0, 0) (0, 1, 1) (1, 0, 2) (1, 1, 3)
a) Find a vector perpendicular to this parallelogram.
b) Use vector methods to find the area of this parallelogram.
4. Use the dot product to find the cosine of the angle between the diagonal of a cube and one of
its edges.
5. Let L be the line that passes through the points (0, −
√
3 , −1) and (0,
√
3 , 1). Let θ be the
angle between L and the vector u⃗ = 1√
2
⟨0, 1, 1⟩. Calculate θ (to the nearest degree).
Assignment 3. Lines and Planes
Read 12.5
You should be able to do the following problems:
Section 12.5/Problems 1 - 58
Hand in the following problems:
1a. Find the equation of the line that passes through (0, 0, 1) and (1, 0, 2).
b. Find the equation of the plane that passes through (1, 0, 0) and is perpendicular to the line in
part (a).
2. The following equation describes a straight line:
r⃗(t) = ⟨1, 1, 0⟩ + t⟨0, 2, 1⟩
a. Find the angle between the given line and the vector u⃗ = ⟨1, −1, 2⟩.
b. Find the equation of the plane that passes through the point (0, 0, 4) and is perpendicular to
the given line.
3. The following two lines intersect at the point (1, 4, 4)
r⃗ = ⟨1, 4, 4⟩ + t⟨0, 1, 0⟩ r⃗ = ⟨1, 4, 4⟩ + t⟨3, 5, 4⟩
a. Find the angle between the two lines.
b. Find the equation of the plane that contains every point o ...
Calculus IDirections (10 pts. each) Answer each of the followin.docxclairbycraft
Calculus I
Directions: (10 pts. each) Answer each of the following questions below. In order to receive ANY credit for a question, you must SHOW YOUR WORK using proper notation and clear and concise logic. You're graded on both the accuracy of your answers AND your explanations that sufficiently support your answers. Unless otherwise stated, you're to give the EXAXCT VALUES of answers instead of decimal approximations. In order to receive ANY credit for any applied/word problem (i.e. Problems #29 - ), you MUST declare a variable (unless the variable(s) have already been declared in the problem) and set up and solve an appropriate mathematical expression that can be used to answer the question. Proper units must also be included in answers to applied problems. NO CREDIT WILL BE GIVEN FOR EITHER GUESSING OR CHECKING POSSIBLE ANSWERS WITHOUT SOLVING THE PROBLEM. YOU CANNOT USE CALCULUS TO SOLVE THESE PROBLEMS.
Finally, write ONLY FINAL ANSWERS ON THESE PAGES; you must show your work both according to homework guidelines and on YOUR OWN PAPER.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Multiply or divide as indicated. Write your answer in factored form.
1) x22 - 9x + 14 · xx22 -- 1618x x ++ 4877 1)
2)
x
-
12
x
+
32
Simplify the complex rational expression.
4
x
2
-
4
x
-
32
-
1
x
-
8
2)
1 + 1 x + 4
Find the difference quotient for the function and simplify it.
3) g(x) = 6x2 + 14x - 1 3)
Find the domain and range of the function. Write your answers using interval notation.
4)
g(z)
=
16
-
z
2
4)
Find a formula for the function graphed.
5) 5)
Determine if the function is even, odd, or neither. You must use algebra to justify your answer; otherwise, no full credit will be given. NO CREDIT is given for an answer without a mathematical explanation.
6) f(x) = x -+7 9 6)
State the domain of the composition.
7)
(
g
H
h)(x) with g(x)
=
x
+
5
and h(x)
=
8
x
+
7
7)
Compute
f(x
+
h)
-
f(x)
h
(h
J
0) for the given function
.
8) f(x) = 4x - 8 8)
9)
f(x)
=
5
x
2
+
6
x
9)
10)
f(x)
=
1
9
x
10)
Solve the equation by multiplying both sides by the LCD.
11) 32x - x 3+ 1 = 1 11)
12)
Solve the equation.
x
+
6
+
2
-
x
=
4
12)
13)
(
4
x
-
2
)
/
3
2
+
6
=
15
13)
14)
3
x
+
4
=
x
-
1
14)
Find the real solutions of the equation by factoring.
15) x3 + 8x2 - x - 8 = 0 15)
Solve the equation by making an appropriate substitution.
16) (x2 - 2x)2 - 11(x2 - 2x) + 24 = 0 16)
Solve the logarithmic equation.
17) log2(x + 7) + log2(x - 7) = 2 17)
Solve the exponential equation. Express the solution set in terms of natural logarithms.
18) 4x + 4 = 52x + 5 18)
Solve the inequality and express the solution in interval notation.
19) 7Ax - 1A L 2 19)
Solve the inequality. Write your answer using interval notation.
20) x 18- 5 > x 15+ 1 20)
Write the equation as f(x) = a(x - h)2 + k. Identify the vertex, range, and axis of symmetry of the function.
21) f(x) = x2 + 5x + 2 21)
23) log
F.
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?
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
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?
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?
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?
48.
49. Further Read
Application of Integration Examples
Application of Integration FAQs
Application of Integration Worksheets
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