How to find the volume under a surface?

In summary, the conversation discusses how the equation z = $$\sqrt{1-x^2}$$ can represent a surface in three dimensions, despite only having two parameters x and y. It is shown that the equation can be rewritten as a circle in the xz-plane, which extends to a cylinder with axis along the y-axis. The speaker also clarifies that x = 0 represents the entire y-z plane, not just the y-axis.
Homework Statement
Find the volume under the surface z = \sqrt{1-x^2} and above the triangle formed y = x , x=1 , and the x-axis.
Relevant Equations
Volume = double integral of the surface function.
I want to know that how can z=$$\sqrt{1-x^2}$$ ever represent a surface? It graphs a curve in the x-z plane and the triangle lies in x-y plane so how can they contain a volume, they are orthogonal to each other. I have attached awn image which is drawn GeoGebra for the function z=$$\sqrt{1-x^2}$$. My question is if we can write the equation z = $$\sqrt{1-x^2}$$ as $$\sqrt{1-x^2} - 0.y - z$$. Then how can it traverse in the y-direction, it's coordinate always have to be zero.
Any help would be appreciated. Thank you.

Attachments

• Screen Shot 2019-08-14 at 9.58.26 PM.png
40.8 KB · Views: 247
You are making a large error when you say "it's coordinate always have to be zero". No, 0*y= 0 for any value of y- that does not mean y= 0!

First, we are talking about three dimensions so "$z= \sqrt{1- x^2}$" means the set of points $(x, y, z)= (x, y, \sqrt{1- x^2})$. There are two "parameters", x and y, so this is a two dimensional surface.

You should be able to see that $z= \sqrt{1- x^2}$ can be written $z^2= 1- x^2$ or $x^2+ z^2= 1$. That is a circle in the xz-plane. Since y can be anything (NOT only 0) that extends to the cylinder with axis along the y-axis. Since we are taking the positive square root the figure is the upper half of the cylinder.

HallsofIvy said:
You are making a large error when you say "it's coordinate always have to be zero". No, 0*y= 0 for any value of y- that does not mean y= 0!

First, we are talking about three dimensions so "$z= \sqrt{1- x^2}$" means the set of points $(x, y, z)= (x, y, \sqrt{1- x^2})$. There are two "parameters", x and y, so this is a two dimensional surface.

You should be able to see that $z= \sqrt{1- x^2}$ can be written $z^2= 1- x^2$ or $x^2+ z^2= 1$. That is a circle in the xz-plane. Since y can be anything (NOT only 0) that extends to the cylinder with axis along the y-axis. Since we are taking the positive square root the figure is the upper half of the cylinder.
Thank you so much. You have made a very nice remark, I have understood, just $x= 0$ doesn’t represent only one point but whole of y-axis.
Thank you so much.

You have made a very nice remark, I have understood, just $x= 0$ doesn’t represent only one point but whole of y-axis.
No. The equation x = 0 represents the whole y-z plane in ##\mathbb R^3##, not just the y-axis.

1. How do I calculate the volume under a surface?

The volume under a surface can be calculated by using the integral calculus method. This involves breaking the surface into small, manageable pieces and calculating the volume of each piece. The sum of all the small volumes will give you the total volume under the surface.

2. What is the formula for finding the volume under a surface?

The formula for finding the volume under a surface is the integral of the surface function over the given bounds. It can be written as V = ∫f(x,y) dA, where f(x,y) is the surface function and dA represents the differential area element.

3. Can I use any surface function to find the volume under a surface?

Yes, you can use any continuous surface function to find the volume under a surface. However, for the integral to be solvable, the function should be defined over the given bounds and have a well-defined area under it.

4. What are the steps involved in finding the volume under a surface?

The steps involved in finding the volume under a surface are: 1) Break the surface into small, manageable pieces. 2) Calculate the volume of each piece using the formula V = ∫f(x,y) dA. 3) Sum up all the small volumes to get the total volume under the surface.

5. Can I use a computer program to find the volume under a surface?

Yes, you can use a computer program such as MATLAB or Wolfram Alpha to find the volume under a surface. These programs use numerical integration techniques to approximate the volume under the surface, which may be helpful for more complex surfaces or when the integral cannot be solved analytically.

Replies
14
Views
1K
Replies
6
Views
2K
Replies
11
Views
2K
Replies
1
Views
749
Replies
4
Views
1K
Replies
8
Views
1K
Replies
1
Views
853
Replies
8
Views
3K
Replies
34
Views
2K
Replies
9
Views
1K