Thread: calculus II areas about axis, pressure, infinite series, MacLaurin series

7.2)
Find the volume formed by rotating the region enclosed by:
and with
about the -axis
7.3)
1. The result of rotating the line about the -axis is
2. The result of rotating the line about the -axis is
3. The result of rotating the line about the line is
4. The result of rotating the line about the line is
5. The result of rotating the line about the line is
6. The result of rotating the line about the line is
7. The result of rotating the line about the line
8. The result of rotating the line about the line

A. a cylinder of radius and height
B. an annulus with inner radius and outer radius
C. an annulus with inner radius and outer radius
D. an annulus with inner radius and outer radius
E. an annulus with inner radius and outer radius is
F. an annulus with inner radius and outer radius
G. a cylinder of radius and height
H. a cylinder of radius and height
7.5)
The region between the graphs of and is rotated around the line .
The volume of the resulting solid is
7.6)
The region between the graphs of and is rotated around the line .
The volume of the resulting solid is
7.7)
A ball of radius 12 has a round hole of radius 4 drilled through its center. Find the volume of the resulting solid.
7.9)
Find the volume of the solid formed by rotating the region enclosed by about the x-axis

7.10)
Find the volume of the solid formed by rotating the region enclosed by

about the -axis.
8.2)
Consider the parametric equation

What is the length of the curve for to ?
8.8)
A trough is 5 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of from to . The trough is full of water. Find the amount of work in foot-pounds required to empty the trough by pumping the water over the top. Note: The weight of water is pounds per cubic foot.

8.9)
A circular swimming pool has a diameter of 16 m, the sides are 3 m high, and the depth of the water is 2.5 m. How much work (in Joules) is required to pump all of the water over the side? (The acceleration due to gravity is 9.8 and the density of water is 1000 .)
8.12)
Find the area of the surface obtained by rotating the curve

from to about the -axis.
8.13)
Find the area of the surface obtained by rotating the curve

from to about the -axis.
8.14)
An aquarium m long, m wide, and m deep is full of water. Find the following:
the hydrostatic pressure on the bottom of the aquarium ,
the hydrostatic force on the bottom of the aquarium ,
the hydrostatic force on one end of the aquarium .
11.1)

For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise.

(a) The series .
(b) The sequence .
11.2)

Determine:
(a) whether is convergent.
(b) whether is convergent.
If convergent, enter the limit of convergence. If not, enter "DIV" (unquoted).
11.6)
A ball drops from a height of 24 feet. Each time it hits the ground, it bounces up 80 percents of the height it fall. Assume it goes on forever, find the total distance it travels.
11.11)
The three series , , and have terms

Use the Limit Comparison Test to compare the following series to any of the above series. For each of the series below, you must enter two letters. The first is the letter (A,B, or C) of the series above that it can be legally compared to with the Limit Comparison Test. The second is C if the given series converges, or D if it diverges. So for instance, if you believe the series converges and can be compared with series C above, you would enter CC; or if you believe it diverges and can be compared with series A, you would enter AD.


1.
2.
3.
11.12)
Match each of the following with the correct statement.
A. The series is absolutely convergent.
C. The series converges, but is not absolutely convergent.
D. The series diverges.

1.
2.
3.
4.
5.
6.
11.13)
Consider the series where

In this problem you must attempt to use the Ratio Test to decide whether the series converges.

Compute

Enter the numerical value of the limit L if it converges, INF if it diverges to infinity, MINF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity.


Which of the following statements is true?
A. The Ratio Test says that the series converges absolutely.
B. The Ratio Test says that the series diverges.
C. The Ratio Test says that the series converges conditionally.
D. The Ratio Test is inconclusive, but the series converges absolutely by another test or tests.
E. The Ratio Test is inconclusive, but the series diverges by another test or tests.
F. The Ratio Test is inconclusive, but the series converges conditionally by another test or tests.
Enter the letter for your choice here:
11.14)
Find all the values of x such that the given series would converge.

The series is convergent
from = , left end included (enter Y or N):
to = , right end included (enter Y or N):
11.15)
Match each of the power series with its interval of convergence.

1.
2.
3.
4.

A.
B.
C.
D.

11.16)
Compute the 9th derivative of

at .


Hint: Use the MacLaurin series for .
11.18)
Assume that equals its Maclaurin series for all x.
Use the Maclaurin series for to evaluate the integral

. Your answer will be an infinite series. Use the first two terms to estimate its value.

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