This cheat sheet covers how calculus is used to find the balance point of objects, regions, curves, and solids. Students need these tools to connect integration with physical quantities like mass, moment, area, and volume. It is especially useful when solving problems involving variable density, symmetry, and rotation about an axis.
Key Facts
- For point masses on a line, the center of mass is .
- For point masses in the plane, the center of mass is and , where .
- For a thin rod on with density , mass is and center of mass is .
- For a plane region with constant density, the centroid is and .
- For a region between and on , area is .
- For a region between and , the moments are and .
- Pappus's centroid theorem for volume states that , where is the distance from the centroid of the plane region to the axis of rotation.
- Pappus's centroid theorem for surface area states that , where is arc length and is the distance from the curve's centroid to the axis of rotation.
Vocabulary
- Center of mass
- The balance point of a system, found by dividing total moment by total mass.
- Centroid
- The geometric center of a region or curve when density is constant.
- Moment
- A measure of rotational tendency, usually computed as distance times mass, area, or density.
- Density function
- A function such as that describes how mass is distributed along an object.
- Pappus's theorem
- A theorem that finds volume or surface area by multiplying a centroid's circular path length by area or arc length.
- Axis of rotation
- The line around which a region or curve is rotated to form a solid or surface.
Common Mistakes to Avoid
- Using area when mass is required is wrong because variable density changes the balance point. Use or when density is not constant.
- Forgetting to divide by total mass or area gives a moment, not a center coordinate. A center coordinate must have the form .
- Using the wrong distance in Pappus's theorem gives an incorrect circular path. The value must be the perpendicular distance from the centroid to the axis of rotation.
- Applying Pappus's theorem when the axis crosses the region or curve is wrong in standard use. The axis of rotation must not intersect the interior of the rotating shape.
- Reversing top and bottom functions can make area negative. For vertical slices, use .
Practice Questions
- 1 Point masses , , and are located at , , and . Find .
- 2 Find the centroid of the region under above the -axis on .
- 3 A region has area and centroid units from an external axis of rotation. Use Pappus's theorem to find the volume formed by rotating the region about that axis.
- 4 Explain why symmetry can sometimes determine a centroid coordinate without evaluating an integral.
Understanding Center of Mass & Pappus's Theorem Reference
A moment measures how strongly mass is placed away from a chosen axis. Distance matters because a small amount of mass far from an axis can balance a larger amount nearby. This is the same idea used on a seesaw, a balance scale, and a wrench.
Calculus handles objects made of countless tiny pieces. Each piece contributes its mass multiplied by its distance. Integration adds those tiny turning effects.
Dividing the total moment by the total mass gives a location that acts like the object's balancing point. The center of mass can lie in empty space. For example, the balance point of a ring is at its center, even though there is no material there.
Density tells the calculation what kind of object is being modeled. A linear density describes mass along a wire or rod. An area density describes a flat sheet.
A volume density describes a three dimensional object. Constant density lets area stand in for mass when finding the centroid of a flat region. Variable density does not allow that shortcut.
A thicker or heavier section pulls the center of mass toward itself. Students should identify the density type before choosing an integral. Units provide a useful check.
Linear density has mass per length, so integrating it over length must give mass. A moment has mass times length. A final center coordinate must have units of length.
For regions bounded by curves, vertical slices are often convenient, but they are not automatic. The upper curve minus the lower curve gives a slice height only when the region is described cleanly from left to right. Some shapes require horizontal slices because a vertical slice would cross different boundaries.
Sketching the region first prevents many errors. The names of the moments can feel backwards. The moment about the vertical axis helps locate the horizontal coordinate, because it records horizontal distance.
The moment about the horizontal axis helps locate the vertical coordinate. Symmetry can save work.
If a uniform shape has mirror symmetry across an axis, its centroid lies on that axis. Symmetry does not determine the other coordinate unless another symmetry is present.
Pappus's theorems offer a fast route for certain solids or surfaces made by rotation. Instead of building many disks or shells, the theorem tracks the path of a centroid during one full turn. The centroid travels a circular distance equal to two pi times its distance from the rotation axis.
Multiplying that travel distance by the original area gives volume. For a rotating curve, multiplying the same travel distance by curve length gives surface area. These theorems require the axis of rotation to stay outside the region or curve being rotated.
If the axis cuts through it, parts overlap and the simple centroid path argument fails. In practice, first find or recognize the centroid, then confirm the axis condition before using Pappus.