How to find the volume under a surface?

[Adesh]

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.

 

[HallsofIvy]

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.

 

[Adesh]

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.

 

[Mark44]

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.