f(x)=3 a(x) xm125/allworksheets125.pdfcompute the volume of the solid generated by rotating the part...

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x 3 1 2 3 4 f(x)=3 A(x) g(x)=1+x 1 2 3 4 1 2 3 B(x) x 4 2 h(x)=6-x 6 x 4 6 2 C(x)

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Page 1: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

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Page 3: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

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Page 4: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

Antiderivatives and Areas Math 125

Name Quiz Section

In this worksheet, we explore the Fundamental Theorem of Calculus and applications of the AreaProblem to problems involving distance and velocity. We also consider integrals involving net andtotal change.

FTC Practice

1 Let f(x) be given by the graph to the right and define

A(x) =∫

x

0

f(t) dt. Compute the following.

A(1) = A(2) =

A(3) = A(4) =

A′(1) = A′(2) =

A′(3) = A′(4) =

The maximum value of A(x) on the interval [0, 5] is

The maximum value of A′(x) on the interval [0, 5] is

1

2

3

21 3 4 5

y=f(x)

Velocity and Distance

2 A toy car is travelling on a straight track. Its velocityv(t), in m/sec, be given by the graph to the right. Define s(t)to be the position of the car in meters. Choose coordinatesso that s(0) = 0. Compute the following.

s(2) = s(4) = s(6) =

v(2) = v(4) = v(6) =

The maximum value of s(t) on the interval [0, 7] is

The minimum value of s(t) on the interval [0, 7] is

The maximum value of v(t) on the interval [0, 7] is

The minimum value of v(t) on the interval [0, 7] is

y=v(t)

1

2

3

-1

2 3 4 5 6 71

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Net and Total Change

3 (a) Evaluate∫

2

−2

∣x2− 4

∣ dx and

2

−2

(

x2− 4

)

dx

and explain your answers.

(b) Now evaluate∫

3

−3

∣x2− 4

∣ dx and∣

3

−3

(

x2− 4

)

dx

and explain your answers.

Page 6: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

Another Area Problem

4 An artist you know wants to make a figure consistingof the region between the curve y = x2 and the x-axis for0 ≤ x ≤ 3

(i) Where should the artist divide the region with a verticalline so that each piece has the same area? (See the picture.)

(ii) Where should the artist divide the region with verticallines to get 3 pieces with equal areas?

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����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

2y=x

4

6

8

2

10

0

1 2 3

Page 7: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

Area Between Curves Math 125

Name Quiz Section

In this work sheet we’ll study the problem of finding the area of a region bounded by curves. We’llfirst estimate an area given numerical information. The we’ll use calculus to find the area of a morecomplicated region.

The Lake

1 The widths, in feet, of a small lake were measuredat 40 foot intervals. Estimate the area of the lake.

70 f

eet

40 feet

45 f

eet

65 f

eet

50 f

eet

80 f

eet

Area Bounded by Three Curves

2 On the grid below sketch the graphs of y = 4, y = x2 and y =

27x. (The last one is just apiece of a sideways parabola).

0 1

16

2 3 4

4

8

12

3 Shade the “triangular” region bounded by the graphs of the three functions that lies above thehorizontal line.

Page 8: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

4 Compute the x-coordinate of the left endpoint of the region.

5 Compute the x-coordinate of the right endpoint of the region.

6 Note that the top of the region consists of a single curve, but the bottom of the region consistsof two different curves. Find the x-coordinate where these two curves meet.

7 Sketch in a vertical line at the x-coordinate you found in the last problem. This divides theregion into two smaller sub-regions.

8 Compute the area of the left sub-region.

Page 9: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

9 Compute the area of the right sub-region. Add the two areas together to get the total area.

10 Recompute the area using the following trick. Solve for x as a function of y in the twonon-constant functions. Find the area by integrating with respect to y. Is this easier?

Page 10: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

Solids of Revolution and the Astroid Math 125

Name Quiz Section

In this worksheet we are going to practice computing somemore volumes of solids of revolution. These will all be basedon a curve called the “astroid”. This curve is formed by rollinga small wheel around the inside of a larger one (see the picture).If the radius of the small wheel is one quarter the radius of thebig one, a point P on the small wheel will trace out the fourpointed curve shown on the far right. It’s called the astroidbecause it looks like a star.

P

1 If the radius of the big wheel is taken to be one, the astroidcan be shown to have the equation x2/3 + y2/3 = 1. Use disks tocompute the volume of the solid generated by rotating the part ofthe astroid in the first quadrant around the y-axis.

1

1

0

Page 11: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

2 Use cylindrical shells to compute the volume of the solid generated by rotating the first quadrantportion of the astroid about the x-axis. (Hint: Try the substitution u3 = y2, so 3u2 du = 2y dy.)How does this compare with your answer in Problem 1? Can you explain this geometrically?

3 Use any method you wish to compute the volumes of the solids generated by rotating the firstquadrant portion of the astroid about the lines x = 1 and y = −1. Set up only. Do not compute

the integrals.

Page 12: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

Integration by Parts Math 125

Name Quiz Section

In this work sheet we’ll study the technique of integration by parts. Recall that the basic formulalooks like this:

∫u dv = u · v −

∫v du

1 First a warm-up problem. Consider the integral∫

x sin(3x) dx. Let u = x and let dv =

sin(3x) dx. Compute du by differentiating and v by integrating, and use the basic formula tocompute the original integral. Don’t forget the arbitrary constant!

2 Compute∫

ln x dx. (The proper technique is, indeed, integration by parts. What should you

take to be u and dv? The choices are pretty limited. Try one and see what happens.)

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3 The regions A and B in the figure are re-volved around the x-axis to form two solids ofrevolution.

(a) Before computing the integrals, which soliddo you think has a larger volume? Why?

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A

y=ln(x)1

3 4 51 2 e

B

(b) Use the disk method to find the volume of the solid swept out by region A.

(c) Use the shell method to find the volume of the solid swept out by region B.

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4 Suppose we try to integrate 1/x by parts, taking u = 1/x and dv = dx. We have du =(−1/x2) dx and v = x, so

∫1

xdx =

1

x· x −

∫x ·

−1

x2dx

= 1 +∫ 1

xdx.

Canceling the integral from both sides, we get the disconcerting result that 0 = 1. What went

wrong? What happens if we replace the indefinite integrals by definite integrals, that is, if we try

to calculate∫

b

a

1

xdx by this method?

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Algebra and Partial Fractions Math 125

Name Quiz Section

Integration of rational functions is mostly a matter of algebraic manipulation. In this worksheet weshall work through some examples of the necessary techniques.

1a Consider the rational function f(x) =2x3

− 4x2− 5x + 3

x2− 2x− 3

. Use long division to get a quotient

and a remainder, then write f(x) =quotient + (remainder/divisor).

1b Now consider the expressionx + 3

x2− 2x− 3

. Factor the denominator into two linear terms.

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1c We wish to writex + 3

(x− 3)(x + 1)as a sum

A

x− 3+

B

x + 1. Let’s find A and B. Set the two

expressions equal and clear denominators (that is, multiply through by (x − 3)(x + 1) and cancel(x − 3)’s and (x + 1)’s as much as possible). Plug in x = 3 and solve for A. Use the same idea tofind B. Check your work by adding the two fractions together.

1d Now use the results of Problems 1a, 1b, and 1c to compute∫ 2x3

− 4x2− 5x + 3

x2− 2x− 3

dx.

1e Some of the terms in the answer to Problem 1d involve logarithms. Combine those terms intoa single term of the form ln(some function of x).

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2a Next, consider the function f(x) =3x + 1

x(x + 1)2. The problem here is that one of the linear

factors in the denominator is squared. Partial fraction theory says the best we can do is to get this

one in the formA

x+

B

x + 1+

C

(x + 1)2. Let’s find A, B and C.

First set the two expressions equal and clear denominators. Plug in x = 0 and solve for A. Plug inx = −1 and solve for C.

2b Now that you’ve found A and C, you can find B by plugging in any other convenient valuefor x. Do so.

2c Now compute∫ 3x + 1

x(x + 1)2dx.

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Techniques of Integration Math 125

Name Quiz Section

The following integrals are more challenging than the basic ones we’ve seen in the textbook so far.You will probably have to use more than one technique to solve them. Don’t hesitate to ask forhints if you get stuck.

1.

∫ sin(t) cos(t)

sin2(t) + 6 sin(t) + 8dt

2.

(

sin−1(x))2

dx

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3.

y2

(1 − y2)7/2dy

4.

∫ 1

x + 2√

x + 1dx

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Differential Equations Math 125

Name Quiz Section

This worksheet walks you through a couple of non-trivial applications of Differential Equations.

Forensic Mathematics

A detective discovers a murder victim in a hotel room at 9:00am one morning. The temperature ofthe body is 80.0◦F. One hour later, at 10:00am, the body has cooled to 75.0◦F. The room is keptat a constant temperature of 70.0◦F. Assume that the victim had a normal temperature of 98.6◦Fat the time of death. We’ll use differential equations to find the time the murder took place.

Let u(t) be the temperature of the body after t hours. By Newton’s Law of Cooling we have thedifferential equation

du

dt= k (u − 70)

where k is a constant (to be determined). We’ll solve the differential equation and get a formulafor u(t).

1 Multiply both sides by dt to get a differential form of the equation. Now do some easy algebrato get the variable u on the same side as the du. (Leave the k where it is).

2 Integrate both sides of the equation. Integrate the right side with respect to t and the left withrespect to u. You can combine the integration constants into one “+C” on the right side.

3 Solve for u as a function of t. Your function will involve the constants k and C.

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4 Take t = 0 when the body was found at 9:00am. Plug in t = 0 and u = 80.0◦F and solve forC. (It’s easier to solve for A = eC and use this in your formula).

5 Plug in t = 1 and u = 75.0◦F and solve for k. (This’ll take some log tricks).

6 Set u = 98.6◦F and solve for t. At what time did the murder take place?

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Spread of a Rumor

The Xylocom Company has 1000 employees. On Monday a rumor began to spread among themthat the CEO had suddenly moved to Brazil. It is reasonable to assume that the rate of the spreadof the rumor is proportional to the number of possible encounters between employees who haveheard the rumor and those who have not. Let y = y(t) be number of employees who have heardthe rumor after t days.

1 Explain why the number of possible meetings between employees who have heard the rumorand those who have not equals y (1000 − y).

2 Write a differential equation that describes this model of the spread of a rumor. (Rememberthat “is proportional to” means “is some constant k times”.)

3 Proceed as in the cooling body problem to solve the differential equation for y(t). You will needto use the method of partial fractions. Your answer should involve two constants: the proportionalityconstant k and a constant C from integrating. (As in part 2 of the previous problem, you cancombine the integration constants into one “+C” on the right side.)

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4 At the very beginning, 50 people had heard the rumor (they all attended the same meeting).Compute the constant A = eC .

5 On Tuesday morning, 100 people had heard the rumor. Compute the constant k.

6 When will 800 people have heard the rumor?

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Review for the Final Exam - Part 2 Math 125

Name Quiz Section

The following problems should help you review for the final exam. Don’t hesitate to ask for hintsif you get stuck.

Arclength and Approximation

1. Write an integral that computes the arclength of the curve y = ex/2 between x = 0 and x = 2.Use Simpson’s Rule with n = 4 subintervals to estimate the value of the integral.

Center of Mass2. Find the center of mass of a plate with constant densitythat occupies the region −

1

4π ≤ x ≤ 0, 0 ≤ y ≤ sec2 x.

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2

1.5

-0.4-0.8

Page 30: f(x)=3 A(x) xm125/AllWorksheets125.pdfcompute the volume of the solid generated by rotating the part of the astroid in the rst quadrant around the y-axis. 1 1 0 2 Usecylindrical shellstocomputethevolumeofthesolidgeneratedbyrotatingthe

Net and Total Distance

3. You throw a ball straight up into the air with velocity 40 ft/sec and catch it (at the sameheight) when it comes back down. What is the total distance traveled by the ball?

Differential Equations

4. Let f(t) be a continuous function and let a be a constant. Show that y = e−at∫ t

0

easf(s) ds

satisfies the differential equationdy

dt+ ay = f(t).

5. An electric circuit with resistance 10 ohms and inductance 2 henrys is powered by a 12-voltbattery. The current I (in amperes) at time t (in seconds) in such a circuit satisfies the differentialequation

2dI

dt+ 10I = 12.

Suppose that I = 0 when the circuit is activated at time t = 0.(a) Find the current I at all times t > 0.(b) Find the limiting value of I as t → ∞.(c) After what time is the current within 0.1 ampere of its limiting value?