Find the Volume of Spheres using Integral Calculus

Hi friends! This video is about deriving and finding the volume of a sphere by using integral calculus!

The formula for the volume of any shape is the definite integral from one end of the shape (let's call it "a") to the other end of the shape (let's call this "b"). So the integral is from a to b of the area of the cross-section (If you cut or slice the shape, you find the area of the open surface) and then dx.

Here, the area of the cross-section of the sphere is the circle! The area of a circle is pi times the radius squared. From the figure shown, the radius is y. Using the Pythagorean theorem, we get y = (r^2-x^2)^0.5. Plugging this into the area of the circle, we get A = pi(r^2-x^2). Now, we substitute this into the integral from -r to r and solve for it. Done.

In the next video, I will show you how to find the volume of a cone using integration.

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[ptyptypty]

Hi Quoc.. your videos are PERFECT... I keep saying that in other videos that you have made. You keep it simple and SO CLEAR!! You are a Genius in your ability to present Calculus videos. You're going to go places with your Intuition and Talent for Teaching. Thank you!

[physicslover]

Bro you are definitely a very great mind

[AtoZPowerwashing]

Hey Quoc, how come at 3:30 the integration of r^2 isn't r^3/3 based on integration rules

[Martin-hiwx]

Great. So no need for using axes x, y and z.?

[qualquan]

why not just calculate for a hemisphere and double it?
prevents dealing with negative quantities