Volume of a Sphere: Solve with Calculus & Integration

[Feodalherren]

Homework Statement

Show that the volume of a sphere of radius r is

V = (4/3)πr^2

Homework Equations

calculus, integration

The Attempt at a Solution

I have the solution in the book but it's confusing me, I'll attach a picture.

So I get lost where it starts talking about a cross-sectional area all of a sudden multiplying by ∏. What's going on here?!

 

[D H]

What's the area of a circle of radius r? What if the radius is y instead of r?

 

[Mark44]

Homework Statement

Show that the volume of a sphere of radius r is

V = (4/3)πr^2

Probably a typo - the volume is (4/3)$\pi r^3$.

Homework Equations

calculus, integration

The Attempt at a Solution

I have the solution in the book but it's confusing me, I'll attach a picture.

So I get lost where it starts talking about a cross-sectional area all of a sudden multiplying by ∏. What's going on here?!

The book is using disks that are oriented vertically (the x-axis intersects each disk perpendicularly). The volume of such a disk is its area times its thickness, dx.