Surface integral from vector calculus

So, to sum up the conversation, the problem involves finding the double integral of ((x2)i + (y2)j + zk) dot product with (-(x/(x2+y2)^ (1/2)i -(y/(x2+y2)^ (1/2)j + (x2 + y2) ^ (1/2)k) over a cone with z = (x2+y2) ^ (1/2), with limits x2 + y2 <= 1, x >= 0, y >= 0, and oriented upward, and after simplifying and using polar coordinates, the result obtained by one student is 0, while others claim it to be (2/3)
  • #1
m453438
6
0

Homework Statement


F = (x^2) i + (y^2) j + (z) k, S is the cone z = (x^2+y^2) ^ (1/2), with x^2 +y^2 <= 1, x >= 0, y >= 0, oriented upward.


Homework Equations


All of the above


The Attempt at a Solution



My attempted solution is 0. But other students claim that the answers is (2/3)pi.

My work is this:
double integral ( (x^2) i + (y^2) j + (z) k ) . ( -(x / (x^2+y^2)^ (1/2) i -(y / (x^2+y^2)^ (1/2) j + (x^2 + y^2) ^ (1/2) k ) dx dy

Then finding common denominator and simplifying:

double integral ( (- x - y ) / (x^2+y^2)^(1/2) ) dx dy

Changing to polar coordinates:

(first integral 0 to 2pi)(second integral 0 to 1) (-cos(theta) - sin(theta) / (r^2) ) * rdrd(theta)

Finally, doing those integrals I have the result of 0.
 
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  • #2
I do not get 0 as the integral.
 
  • #3
Welcome to PF!

Hi m453438! Welcome to PF! :smile:

(have a theta: θ and a pi: π and a √ and a ∫ and a ≤ and ≥, and try using the X2 tag just above the Reply box :wink:)

Useful tip: when you do it again, try putting r = √(x2 + y2) at the start

it makes it a lot easier to write, which means you're less likely to make a mistake. :wink:
 

What is a surface integral?

A surface integral is a mathematical tool used in vector calculus to calculate the flux of a vector field across a given surface. It is also used to find the total surface area of a curved surface.

What is the difference between a surface integral and a line integral?

The main difference between a surface integral and a line integral is that a surface integral is calculated over a two-dimensional surface, while a line integral is calculated over a one-dimensional curve. Additionally, a line integral takes into account the direction of the curve, while a surface integral does not.

How is a surface integral calculated?

A surface integral is calculated by first parameterizing the given surface with two variables, such as u and v. This creates a function where the values of u and v correspond to points on the surface. Then, the vector field is integrated over this function using a double integral, with the bounds being the range of u and v values.

What is the physical interpretation of a surface integral?

The physical interpretation of a surface integral is the total amount of a vector quantity that is flowing through or across a given surface. It can also be thought of as the total amount of "stuff" passing through a surface. This can have applications in fluid dynamics, electromagnetism, and other fields.

What are some real-world applications of surface integrals?

Surface integrals have many real-world applications, such as calculating the flow of fluid through a pipe, determining the flux of an electric field through a charged surface, and finding the surface area of a curved object. They are also used in computer graphics to render 3D images and in physics to calculate the total energy of a system.

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