Math 211

These problems are provided to help you study. The presence of a
problem on this handout does not imply that there *will* be a
similar problem on the test. And the absence of a topic does not
imply that it *won't* appear on the test.

1. Compute the following integrals.

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(i) .

2. Determine whether the integral converges or diverges. Find the value of the integral if it converges.

3. Prove by comparison that converges.

4. The region between the x-axis and from to is revolved about the line . Find the volume generated.

5. Let R be the region bounded above by , bounded below by , and bounded on the sides by and by the y-axis. Find the volume of the solid generated by revolving R about the line .

6. Find the area of the region which lies between the graphs of and , from to .

7. Find the area of the region between and from to .

8. The base of a solid is the region in the x-y-plane bounded above by the curve , below by the x-axis, and on the sides by the lines and . The cross-sections in planes perpendicular to the x-axis are squares with one side in the x-y-plane. Find the volume of the solid.

9. A tank built in the shape of the bottom half of a sphere of radius 2 feet is filled with water. Find the work done in pumping all the water out of the top of the tank.

10. Does the following series converge absolutely, converge conditionally, or diverge?

11. Determine whether the series converges absolutely, converges conditionally, or diverges.

12. Does the series converge absolutely, converge conditionally, or diverge?

13. Find the sum of the series

14. In each case, determine whether the series converges or diverges.

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

15. Find the values of x for which the following series converges absolutely.

16. The series converges by the Alternating Series Test. Determine the smallest value of n for which the partial sum approximates the actual sum to within 0.01.

17. (a) Find the Taylor expansion at for .

(b) Find the Taylor expansion at for . What is the interval of convergence?

18. (a) Use the Binomial Series to write out the first three nonzero terms of the series for .

(b) Find the first three terms of the Taylor series at for by integrating the series you got in (a) from to .

19. Use the Remainder Term to find the minimum number of terms of the Taylor series at for needed to approximate on the interval to within .

20. Determine the interval of convergence of the power series .

21. Find the interval of convergence of the power series .

22. If and , find and at .

23. Consider the parametric curve

(a) Find the equation of the tangent line at .

(b) Find at .

24. Find the length of the loop of the curve

25. Find the length of the curve for .

26. Let

Find the length of the arc of the curve from to .

27. Find the area of the surface generated by revolving , , about the x-axis.

28. Find the area of the surface generated by revolving , , about the x-axis.

29. (a) Convert to polar and simplify.

(b) Convert to rectangular and describe the graph.

30. Find the slope of the tangent line to the polar curve at .

31. Find the slope of the tangent line to at .

32. Find the values of in the interval for which the polar curve passes through the origin.

33. Find the length of the cardioid .

34. Find the area of the intersection of the interiors of the circles

35. Find the area of the region inside the cardioid and outside the circle .

36. Let A be the region inside and let B be the region inside . Find the area of the intersection of A and B --- that is, the area of the region common to A and B.

1. Compute the following integrals.

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(i) .

(a)

(b)

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(c)

Setting gives , so .

Setting gives , so .

Therefore,

Setting gives , so .

Thus,

(d)

(e)

(f) I need to complete the square. Note that and . Then

So

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(g)

Let . I get

Then

Let . I get

Then

Let . I get

Thus,

(h)

(i)

Use long division to divide by :

Thus,

2. Determine whether the integral converges or diverges. Find the value of the integral if it converges.

and , so the integral diverges.

3. Prove by comparison that converges.

Since for ,

Since converges, the original integral converges as well.

4. The region between the x-axis and from to is revolved about the line . Find the volume generated.

Use shells. The height of a shell is , and the radius is . The volume is

5. Let R be the region bounded above by , bounded below by , and bounded on the sides by and by the y-axis. Find the volume of the solid generated by revolving R about the line .

Most of the things in the picture are easy to understand --- but why is ?

Notice that the *distance* from the y-axis to the side of the
shell is , not x. Reason: x-values to the left of the y-axis
are *negative*, but distances are always *positive*.
Thus, I must use to get a positive value for the distance.

As usual, r is the distance from the axis of revolution to the side of the shell, which is .

The left-hand cross-section extends from to . You can check that if you plug x's between -2 and 0 into , you get the correct distance from the side of the shell to the axis .

The volume is

6. Find the area of the region which lies between the graphs of and , from to .

As the picture shows, the curves intersect. Find the intersection point:

On the interval , the curves cross at . I'll use vertical rectangles. From to , the top curve is and the bottom curve is . From to , the top curve is and the bottom curve is . The area is

7. Find the area of the region between and from to .

As the picture shows, the curves intersect. Find the intersection point:

I'll use vertical rectangles. From to , the top curve is and the bottom curve is . From to , the top curve is and the bottom curve is . The area is

8. The base of a solid is the region in the x-y-plane bounded above by the curve , below by the x-axis, and on the sides by the lines and . The cross-sections in planes perpendicular to the x-axis are squares with one side in the x-y-plane. Find the volume of the solid.

The volume is

9. A tank built in the shape of the bottom half of a sphere of radius 2 feet is filled with water. Find the work done in pumping all the water out of the top of the tank.

I've drawn the tank in cross-section as a semicircle of radius 2 extending from to .

Divide the volume of water up into circular slices. The radius of a slice is , so the volume of a slice is . The weight of a slice is , where I'm using 62.4 pounds per cubic foot as the density of water.

To pump a slice out of the top of the tank, it must be raised a distance of feet. (The "-" is necessary to make y positive, since y is going from -2 to 0.)

The work done is

10. Does the following series converge absolutely, converge conditionally, or diverge?

The absolute value series is

Note that when n is large,

Hence, I'll compare the series to .

The limit is finite ( ) and positive ( ). The harmonic series diverges. By Limit Comparison, the series diverges. Hence, the original series does not converge absolutely.

Returning to the original series, note that it alternates, and

Let . Then

Therefore, the terms of the series decrease for , and I can apply the Alternating Series Rule to
conclude that the series converges. Since it doesn't converge
absolutely, but it *does* converge, it converges
conditionally.

11. Determine whether the series converges absolutely, converges conditionally, or diverges.

The absolute value series is .

converges because it's a p-series with .

The absolute value series converges by limit comparison.

The original series converges absolutely.

12. Does the series converge absolutely, converge conditionally, or diverge?

Consider the absolute value series .

converges, because it's a p-series with . Hence, the absolute value series converges by Limit Comparison.

Therefore, the original series converges absolutely.

13. Find the sum of the series

14. In each case, determine whether the series converges or diverges.

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(a) Apply the Integral Test. The function is positive and continuous on the interval .

Note that

It follows that for . Hence, f decreases on the interval . The hypotheses of the Integral Test are satisfied.

Compute the integral:

(To do the integral, I substituted , so .)

Since the integral converges, the series converges by the Integral Test.

(b) Apply the Ratio Test. The term of the series is

Hence, the -st term is

Hence,

The limiting ratio is

The limit is less than 1, so the series converges, by the Ratio Test.

(c) Apply the Root Test.

The limit is

Since , the series converges, by the Root Test.

(d) Note that

It follows that is undefined --- the values oscillate, approaching . Since, in particular, the limit is nonzero, the series diverges, by the Zero Limit Test.

(e) Apply Limit Comparison:

The limit is finite and positive. The series diverges, because it's a p-series with . Therefore, the original series diverges by Limit Comparison.

(f)

diverges, because it's 4 times the harmonic series. Therefore, diverges by Direct Comparison.

(g) The series has positive terms.

is continuous for .

Compute the derivative:

for all x, and for . Therefore, for . Hence, decreases for .

The three conditions for applying the Integral Test are satisfied. Compute the integral:

Here's the work for the integral:

Here's the work for the two limits. I used L'Hôpital's Rule to compute the first limit.

Since the integral converges, the series converges by the Integral Test.

(h) This is *not* an alternating series, even though it
contains a !

The series looks like for large k; use Limit Comparison. The limiting ratio is

The limit is nonzero and finite. converges, because it's a p-series with . Therefore, converges by Limit Comparison.

15. Find the values of x for which the following series converges absolutely.

Apply the Ratio Test:

The limiting ratio is

The series converges absolutely for , i.e. for . The series diverges for and for .

You'll probably find it difficult to determine what is happening at the endpoints! However, if you experiment --- compute some terms of the series for , for instance --- you'll see that the individual terms are growing larger, so the series at and at diverge, by the Zero Limit Test.

16. The series converges by the Alternating Series Test. Determine the smallest value of n for which the partial sum approximates the actual sum to within 0.01.

The error in approximating the exact value of the sum by is less than the term, which is . So I want

Take .

17. (a) Find the Taylor expansion at for .

(b) Find the Taylor expansion at for . What is the interval of convergence?

(a)

(b)

The series converges for , i.e. for .

18. (a) Use the Binomial Series to write out the first three nonzero terms of the series for .

(b) Find the first three terms of the Taylor series at for by integrating the series you got in (a) from to .

(a)

(b)

19. Use the Remainder Term to find the minimum number of terms of the Taylor series at for needed to approximate on the interval to within .

First, I'll find the Remainder Term.

Hence, for some z between 0 and x,

I want .

Since , I have

Moreover, since z is between 0 and x and , I also have . So

Therefore,

Therefore, I want the smallest value of n such that

This inequality can't be solved algebraically, due to the factorial in the denominator. So I have to do this by trial and error.

The smallest value of n that works is .

20. Determine the interval of convergence of the power series .

By the Ratio Test, the series converges for . Hence, the base interval is : .

At , the series is . It diverges, because it's harmonic.

At , the series is . It converges, because it's alternating harmonic.

The interval of convergence is .

21. Find the interval of convergence of the power series .

Apply the Ratio Test to the absolute value series:

The series converges for , i.e. for .

At , the series is

It's harmonic, so it diverges.

At , the series is

This is the alternating harmonic series, so it converges.

Therefore, the power series converges for , and diverges elsewhere.

22. If and , find and at .

When , .

When , .

23. Consider the parametric curve

(a) Find the equation of the tangent line at .

(b) Find at .

(a)

When , , , and . The equation of the tangent line is

(b)

When , .

24. Find the length of the loop of the curve

I'll do the easy part first, which is to find the integrand for the arc length. It is

(Note that since , you know that .)

To find the limits of integration, I have to find two values of t which give the same values of x and y. The loop is traced out between these limits.

Note that is the same for t and , because of the square. Note also that , so for and .
Therefore, the values make , *and* since they're negatives of one another
they give the same x-value. In other words, they give the same point
on the curve. Thus, the loop is traced out from to .

The length is

25. Find the length of the curve for .

The length is

Here's the work for the integral:

26. Let

Find the length of the arc of the curve from to .

Hence,

Therefore,

The length is

27. Find the area of the surface generated by revolving , , about the x-axis.

Hence,

Notice that this is just with the sign of the middle term changed. But was squared, so must be squared:

Thus,

The area is

28. Find the area of the surface generated by revolving , , about the x-axis.

The derivative is

The curve is being revolved about the x-axis, so the radius of revolution is . The area of the surface is

29. (a) Convert to polar and simplify.

(b) Convert to rectangular and describe the graph.

(a)

(b)

The graph is a circle of radius centered at .

30. Find the slope of the tangent line to the polar curve at .

When , . Since , when , .

The slope of the tangent line is

31. Find the slope of the tangent line to at .

First, . When , and .

Therefore,

32. Find the values of in the interval for which the polar curve passes through the origin.

Set :

I'll solve . Since the argument of the equation above is , I need solutions in the range . By basic trigonometry,

Thus, the solutions are

Set and solve for :

33. Find the length of the cardioid .

By the double angle formula,

Thus,

The length is

34. Find the area of the intersection of the interiors of the circles

Convert the two equations to polar:

Set the equations equal to solve for the line of intersection:

The region is "orange-slice"-shaped, with the bottom/right half bounded by from to and the top/left half bounded by from to . Hence, the area is

35. Find the area of the region inside the cardioid and outside the circle .

Find the intersection points:

I'll find the area of the shaded region and double it to get the total. The shaded area is

The cardioid area is

The circle area is

Thus, the shaded area is

The total area is .

36. Let A be the region inside and let B be the region inside . Find the area of the intersection of A and B --- that is, the area of the region common to A and B.

Find the intersection point:

(The circles also intersect at the origin, but they pass through the origin at different values of .)

The shaded area is the sum of the area inside from to and the area inside from to :

*The best thing for being sad is to learn something.* -
Merlyn, in T. H. White's *The Once and Future King*

Copyright 2020 by Bruce Ikenaga