MAT1236 Exam II Information

Date and Time 5/9 Thursday

Sections All sections after the first exam

Total Points 60 points

The Chain Rule

1. If , then

2. If , then

3. Other cases.

Fubini’s Theorem

If f is continuous on , then

Splitting Formula

If on , then

General Region D

Type I:

Type II:

Both: Use the order that is easier to compute the integral.

Double Integrals in Polar Coordinates

for

Triple Integral Over a Box

Fubini’s Theorem

Splitting Formula

General Region D Type n:

Cylindrical Coordinates

Spherical Coordinates

Spherical Wedge

Sequences 5 tools you can use to find limits

1. If and , then .

2. The Limit Laws

3. The Squeeze Theorem

4. If , then .

5.

Partial Sum Sequence

Given a series , for , we define its partial sum sequence as

That is, is the sum of the first terms of the series.

Convergence of a Series

If is convergent and then is convergent and . Otherwise is divergent.

Other Concepts

Partial Fractions, Partial Sum Sequence, Telescoping Sum, Index Shifting of Summation

Important Notice

It is extremely important that you use the logical formats and show the necessary details. Answers alone or without proper support arguments will be given no points.

Practice Problems

(Disclaimer: This practice exam has no direct relations with the real exam. You need to understand that the problems in the real exam may not resemble the homework problems or the problems in this practice exam. )

1. Determine whether the series converges or diverges:

Find its sum if it converges.

2. If , where and , show that .

3. Let , where and . If , compute

.

4. Find the volume of the solid under and above region .

5. Find the volume of the solid lies within both and .

6. Evaluate where E lies between and in the first octant.

7. Find the volume of the solid that lies within the sphere , above the xy-plane, and below the cone . Can we use cylindrical coordinates to find the volume?

Answers

1. is convergent and

3. 04.

5. 6.

7. , Yes (But do you know how?)

PPFTNE

(Note that this is not an exhausted list. Do you know what the other problems are?)

1. State and Prove the Splitting Formula for Iterated Integrals.

Splitting Formula
If on , then
Proof

2. State the Squeeze Theorem for sequences.

If and , then .

3. Define the partial sum sequence of a series.

Given a series , for , we define its partial sum sequence as .

4. Define the convergence of a series.

Let be the partial sum sequence of .
If is convergent and then is convergent and . Otherwise is divergent.

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