Derivation of Formula for Volume of the Sphere by Integration

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  • 0:04 Finding the Volume of a Sphere
  • 0:47 Using Integration for Volume
  • 3:33 How to Compare Two Spheres
  • 5:57 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we derive the formula for finding the volume of a sphere. This formula is derived by integrating differential volume elements formed by slicing the sphere into cylinders with a differential thickness.

Finding the Volume of a Sphere

Did you know in some parts of the world, the fruit called an orange is actually green in color even when it has fully ripened? Imagine a perfectly spherical green orange.

The radius of a sphere is R. It's the distance from the center of the sphere to any point on its surface. We can calculate everything we need to know about the sphere from its radius. In particular, we can determine the volume of a sphere, V. The volume is how much is contained inside the sphere. The volume of a sphere is simply:


volume_of_sphere


For the orange, the volume tells us something about the amount of delectable food inside.

Using Integration for Volume

We can derive this equation by slicing up the sphere and integrating. Here are the steps.

Step 1: Take a vertical slice of the sphere.

Imagine slicing a perfect orange or any other sphere vertically. The path of the knife is in blue.


A vertical slice of the sphere
A_vertical_slice


Step 2: Look down the y-axis.

If we position ourselves at some place outside of the sphere and look down the y-axis towards the origin, we see the blue slice as a circle with a radius z.


Looking down the y-axis
Looking_down_the_y_axis


The area of this circle is πz2.

Step 3: Take a side view.

Now we position ourselves outside the sphere along the x-axis. We see the triangle that you're looking at on your screen now.


A side view
A _side_view


This is a right triangle with one side the same green z as before. The hypotenuse is R, the radius of the sphere. The other side of this triangle is y. From the Pythagorean Theorem, z2 + y2 = R2. Solving for z2:


z^2=R^2-y^2


Step 4: Take another vertical slice and define a differential volume, dV.

If we take another vertical slice close to the first slice, we can define a volume.


A differential volume
A_differential_volume


The width of the slice is dy and as a differential element, it's really small - so small, the two slices have the same radius and the same area. In the figure, the size of dy is exaggerated. Otherwise, we wouldn't be able to see two separated slices. It's like we have a cylinder as the volume element. We know the volume of a cylinder is area of base times the height. The area of the base is the area of the circle, πz2 and the height is dy. The volume itself is a differential volume we call dV. Thus,


dV_in_terms_of_z_and_y


Step 5: Integrate dV to find the total volume, V.

Replace dV with πz2 and integrate:


integrate_dV_to_get_V


Replace z2 with R2 - y2 and integrate from y = - R to y = R:


sub_for_z^2_and_integrate_from_-R_to_R


Take the constant π outside of the integral. The anti-derivative of R2 is R2y, and the anti-derivative of y2 is y3/3. Evaluate from y = -R to y = R:


pi_outside_and_anti-derivatives


Substituting the upper limit into the anti-derivative expression and then subtracting the substitution of the lower limit into the anti-derivative expression:


anti-derivatives_and_evaluating


Simplifying:


substituting_limits


Grouping terms, factoring out a 2R3 and simplifying.


simplifying


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