Volume of Revolution Discussion i

The formula for finding a volume of a solid formed by revolving a smooth and continuous function f(x) where a \leq x \leq b, 360° around the x axis is

    \[\pi \int^b_a \big( f(x) \big)^2 dx\]

Let’s find out why. Here’s the outline of the discussion:

  1. Examine the simple case, f(x)=3.
  2. Agree that the function V(x) exists.
  3. Agree that if f(x) is smooth and continuous, then V(x) is smooth and continuous.
  4. Conclude that V'(x)=\pi\big(f(x)\big)^2 which leads to V(x)=\pi \int^b_a\big(f(x)\big)^2dx

Examine a simple case

Let’s take f(x)=3, from x=0 to x=b.

By rotating the straight, horizontal line y=3 around the x axis, we generate a cylinder. We already know that the volume of a cylinder can be found using the formula:

    \[V=\pi r^2 h\]

Where r is the radius of the cross sectional circle, and h is the length or height of the cylinder.

We also remember that a function maps a value from one set (domain) to another set (range). Let’s create a function V that maps the length of the cylinder to the volume of the cylinder. Our domain is the length of the cylinder, which in our case is the measurement on the x axis. If we keep one end of the cylinder parked at 0, then the length of the cylinder is just b.

The cylinder length =b. The cylinder radius =3. Using the volume of cylinder formula given above, our volume function is:

    \[V(b)=\pi \times 3^2 b\]

We see that this agrees with \pi \int^b_0 \big( f(x) \big)^2 dx when f(x)=3 as follows:

    \[\pi \int^{b}_0 \big( 3 \big)^2 dx=\pi\big[9x\big]^{b}_0= \pi \times 9b\]

Let’s check out the volume of the cylinder when a=0 and b=10, using our function V(b):

    \[V(10)=\pi \times 3^2 \times 10=90\pi = 283\text{ (3 s.f.)}\]

Now let’s check out the volume of the cylinder when a=0 and b=4.

    \[V(4)=\pi \times 3^2 \times 4=36\pi = 113\text{ (3 s.f.)}\]

Lastly, let’s check out the volume of the cylinder when a=4 and b=10. We can calculate this by taking the longer cylinder (from zero to ten) and subtracting the shorter cylinder (from zero to 4).

    \[V(10)-V(4)=\pi \times 3^2 \times 10-\pi \times 3^2 \times 4=90 \pi - 36 \pi = 54 \pi\]

The calculation above is exactly the calculation

    \[\pi \int^{10}_4 \big( 3 \big)^2 dx=\pi\big[9x\big]^{10}_4=\pi\big[90-36\big]=54\pi\]

Let’s move on to understand why this calculation works for any smooth, continuous function f(x), not just the simple straight line that generates the familiar cylinder.

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