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

[a0w]

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

 

[HOI]

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?