Calculus 3 Help: Iterated, Double, Triple Integrals & More

In summary: They are all very simple. In summary, the conversation includes topics such as iterated integrals, double and triple integrals, and change of variables in various coordinate systems. These topics include solving problems and understanding concepts from Calculus I and II.
  • #1
a0w
1
0
Kindly help me with:


Iterated integrals
Q1, 2,3
Double integrals in polar coordinates
Q1, 2,3
Triple integrals
Q1, 2,3
Triple integrals in cylindrical coordinates
Q1, 2,3
Triple integrals in spherical coordinates
Q1, 2,3
Change of variables
Q7,8,9
Green's theorem
Q1,2
Surface integrals
Q1,2,3
Divergence theorem
Q1,2,3
Stokes theorem
Q1,2,3
 

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  • #2
Seriously? You post a number of different problems without showing any work of your own? In order to help you we need to know what you understand about the problem, what you can do already and where you have difficulty. You can show us that by showing us what you have attempted to do yourself.

Here, you have posted- I started to count how many problems you posted but then gave up! Many of these are fairly straightforward exercises- most of them are just slight variations on problems you should have seen in Calculus I or Calculus II.

I will take a look at the very first problem: You are asked to find $\int_R\int 16xy- 9x^2+ 1 dA$ where R is $[2, 3]\times [-1, 1]$.

Do you undestand what that means? Do you understand that R is the rectangle in the xy- plane with vertices (2, -1), (3, -1), (2, 1), and (3, 1)? That x goes from 2 to 3 while y goes from -1 to 1? Do you understand that "dA" is shorthand for "dxdy"?

Do you see that the integral is $\int_{y= -1}^1\int_{x= 2}^3 (16xy- 9x^2+ 1) dxdy$$= \int_{y= -1}^1\int_{x= 2}^3 16xy dxdy- 9\int_{y= -1}^1\int_{x= 2}^3 x^2 dxdy+ \int_{y= -1}^1\int_{x= 2}^3 dxdy$$= 16\left(\int_{y= -1}^1ydy\right)\left(\int_{x= 2}^3xdx\right)- 9\left(\int_{y= -1}^1dy\right)\left(\int_{x= 2}^3x^2dx\right)+ \left(\int_{y= -1}^1dy\right)\left(\int_{x= 2}^3dx\right)$.

Can you do those six integrals?
 

FAQ: Calculus 3 Help: Iterated, Double, Triple Integrals & More

1. What is the difference between a single, double, and triple integral?

A single integral is used to find the area under a curve in one dimension. A double integral is used to find the volume under a surface in two dimensions. A triple integral is used to find the hypervolume under a hypersurface in three dimensions.

2. How do I set up an iterated integral?

An iterated integral is used to evaluate a double or triple integral by breaking it down into smaller, single integrals. The innermost integral is evaluated first, and then the result is used in the next integral, and so on until all integrals are evaluated.

3. What is the purpose of using polar, cylindrical, and spherical coordinates in integrals?

Polar, cylindrical, and spherical coordinates are alternative coordinate systems that can make it easier to evaluate certain integrals, particularly those involving circular or spherical shapes. They allow for simpler bounds and easier computation.

4. How do I know which integration technique to use for a given problem?

The integration technique used depends on the type of function being integrated. For example, if the function is a polynomial, the power rule can be used. If the function is trigonometric, trigonometric identities may be used. It is important to be familiar with a variety of integration techniques and to practice using them in different situations.

5. How can I check my answer to an iterated integral?

One way to check your answer is to use a graphing calculator or software to graph the original function and the region of integration. The area or volume should match your calculated answer. You can also try using different integration techniques to see if you get the same result. Additionally, you can check your answer by using the fundamental theorem of calculus to take the derivative of your integral and see if it matches the original function.

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