Problem 10 - The Calculus with Analytic Geometry

Question

Find by integration the volume of a right-circular cone of altitude \(h\) units and base radius \(a\) units.

Step-by-Step Solution

Answer

The volume of the cone is \[ V = \frac{1}{3} \pi a^2 h.\]

Step 1: Set up the Integral

The formula for the volume of a solid of revolution is \[ V = \pi \int_0^h [R(y)]^2 dy\] where R(y) represents the radius of the cone at height y. For a cone, the radius linearly decreases from a at the base to 0 at the tip. Therefore, R(y) is given by \[ R(y) = a \left(1 - \frac{y}{h}\right).\]

Step 2: Simplify the Integral Expression

We substitute R(y) into the volume integral: \[ V = \pi \int_0^h \left(a \left(1 - \frac{y}{h}\right)\right)^2 dy.\] This simplifies to: \[ V = \pi \int_0^h a^2 \left(1 - \frac{y}{h}\right)^2 dy.\]

Step 3: Expand the Inner Expression

Expand the square inside the integral: \[ \left(1 - \frac{y}{h}\right)^2 = 1 - \frac{2y}{h} + \left(\frac{y}{h}\right)^2.\] Therefore, the integral becomes: \[ V = \pi a^2 \int_0^h \left(1 - \frac{2y}{h} + \frac{y^2}{h^2}\right) dy.\]

Step 4: Separate the Integral

The integral can now be separated into three simpler integrals: \[ V = \pi a^2 \left(\int_0^h 1 dy - 2 \int_0^h \frac{y}{h} dy + \int_0^h \frac{y^2}{h^2} dy \right).\]

Step 5: Evaluate Each Integral

Evaluate each of the integrals: \[ \int_0^h 1 dy = y \Big|_0^h = h,\] \[ \int_0^h y dy = \frac{y^2}{2} \Big|_0^h = \frac{h^2}{2},\] and \[ \int_0^h y^2 dy = \frac{y^3}{3} \Big|_0^h = \frac{h^3}{3}.\] Substitute these into the original separated integral: \[ V = \pi a^2 \left(h - 2 \cdot \frac{h^2}{2h} + \frac{h^3}{3h^2} \right).\]

Step 6: Simplify the Expression

Simplify the expression: \[ V = \pi a^2 \left(h - h + \frac{h}{3}\right).\] This simplifies further to \[ V = \pi a^2 \cdot \frac{h}{3}.\]

Step 7: Final Volume Expression

Combine constants to find the final volume: \[ V = \frac{1}{3} \pi a^2 h.\]

Key Concepts

calculus integration

In calculus, integration allows us to find areas, volumes, and other quantities under curves. Think of integration as the reverse process of differentiation. Here, we apply it to find the volume of a cone. We start with a function that describes the cone's dimensions and use integration to sum up infinitely small slices of the cone.
Since we are dealing with a solid shape, we use the definite integral to sum the slices from the bottom to the top of the cone. For our cone, the height goes from 0 to h. The formula for volume using integration is:
\[ V = \pi \int_0^h [R(y)]^2 dy \]
This equation helps us capture the varying radius as the height changes. By integrating, we calculate the total volume of the entire shape.

solid of revolution

A solid of revolution is a 3D shape obtained by rotating a 2D curve around an axis. Imagine taking a right-circular cone's side view, which is a triangle, and rotating it around the vertical axis. This generates the cone.
The key idea is to imagine slicing the cone into many thin discs. Each disc's volume can be added up to find the total volume of the cone. The discs' radii change along the height, starting at the base (radius a) and shrinking to 0 at the tip.
Using the formula for solids of revolution:
\[ V = \pi \int_0^h [R(y)]^2 dy \]
We set up our integral by defining the radius function R(y), which linearly decreases from a to 0.

volume calculation

To find the volume of our cone, we perform a series of steps. First, we define the radius function:
\[ R(y) = a \left(1 - \frac{y}{h}\right). \]
This function tells us how the radius changes with height. We then substitute this function into our volume formula and simplify:
\[ V = \pi \int_0^h [a(1 - y/h)]^2 dy \]
Next, we expand the expression inside the integral and simplify further by splitting it into three simpler integrals. These integrals are:

• \[ \int_0^h 1 dy = h \]
• \[ \int_0^h y dy = \frac{h^2}{2} \]
• \[ \int_0^h y^2 dy = \frac{h^3}{3} \]

Adding these results together and simplifying gives us the final volume formula:
\[ V = \pi a^2 \frac{h}{3} = \frac{1}{3} \pi a^2 h \]
This formula tells us that the volume of a right circular cone depends on both the base radius and height.