The video “Calculus 1 Lecture 5.2: Volume of Solids By Disks and Washers Method” by Professor Leonard explains how to find the volume of three-dimensional solids formed by revolving a region around an axis, using the disks and washers method, which extends the ideas of integration and summation.
Explanation#
1. Volume by Slicing#
- The core concept is similar to finding areas by summing slices, but now slices are thin slabs (disks or washers) perpendicular to an axis, each with a small thickness $\Delta x$.
- The volume of each thin slice can be approximated as the cross-sectional area of the slice times its thickness.
- Summing these volumes and taking the limit as slice thickness approaches zero gives the exact volume:
$$ V = \lim_{n \to \infty} \sum_{k=1}^n A(x_k^*) \Delta x = \int_a^b A(x) , dx $$
where $A(x)$ is the cross-sectional area function.1
2. Cross-Sectional Area#
- Each cross-section is typically a circle or washer (ring) created by revolving a region around an axis.
- The area of a cross-section is $ \pi r^2 $ for a disk (solid circle).
- For washers (disks with holes), the area is:
$$ \pi (R^2 - r^2) $$
where $ R $ is the outer radius and $ r $ is the inner radius (hole radius).1
3. Radius as a Function of x#
- The radius $ r $ or radii $ R, r $ depend on the function describing the curve being revolved.
- Typically, $ r = f(x) $ or $ R = f(x) $, so radius varies along the axis of revolution.
- So cross-sectional area is a function of $ x $, and that is what goes into the integral.
4. Method of Disks#
- When revolving a region that touches the axis of revolution, the slices are disks with no hole.
- You use $ A(x) = \pi [f(x)]^2 $.
- The volume is:
$$ V = \int_a^b \pi [f(x)]^2 , dx $$
5. Method of Washers#
- If the region is bounded away from the axis of revolution, slices have holes, resembling washers.
- Use:
$$ V = \int_a^b \pi \left( [R(x)]^2 - [r(x)]^2 \right) , dx $$
- The integral subtracts the inner radius squared from the outer radius squared.
6. Examples and Setup#
- The video covers:
- How to express radius in terms of $x$,
- When cross-sections are circles,
- How to set up limits $a$ and $b$,
- How to square the function before integrating,
- How to handle constants (e.g., factor out constants like $\pi$),
- Simple examples such as revolving $y = 3\sqrt{x}$ from $x=1$ to $x=4$ around the x-axis resulting in integrals like:
$$ V = \pi \int_1^4 (3\sqrt{x})^2 , dx = 9\pi \int_1^4 x , dx $$
7. Link to Geometry#
- Many solids formed by revolution correspond to well-known shapes like cylinders, cones, and spheres, whose volumes match the integrals computed by this method.
- The slicing method generalizes formulae for volumes by summing infinitely many infinitesimal disks or washers.
8. Application and Continuation#
- This method allows volume calculation for arbitrary curves rotated about axes.
- It is foundational for problems in calculus and physics involving solids of revolution.
- Further lectures cover more complex solids and alternative axes of revolution.
Summary#
Professor Leonard’s lecture teaches the disks and washers method to find volumes of solids generated by revolving curves about an axis. By slicing the solid into thin cross-sectional disks or washers, computing their areas, and integrating, one obtains exact volumes. This method generalizes geometric volume formulas using calculus integration, with radius functions depending on the curves being revolved, and constants factored out to simplify calculations.The video “Calculus 1 Lecture 5.2: Volume of Solids By Disks and Washers Method” by Professor Leonard explains how to compute volumes of solids generated by revolving a region around an axis using integration.
Explanation#
Volume by Slicing#
- Just as area under a curve is found by summing thin rectangles, volume can be approximated by summing thin slabs (disks or washers).
- Each slab’s volume ≈ cross-sectional area × thickness.
- Taking the limit results in the exact volume:
$$ V = \int_a^b A(x) , dx $$
where $A(x)$ is the cross-sectional area expressed as a function of $x$.
Cross-Sectional Area#
- When revolving around the x-axis, each cross-section is a circle with radius equal to the distance from the x-axis to the curve.
- Area of each disk (slice) is $ \pi r^2 $, with $ r = f(x) $.
- For solids with “holes” (washer method), cross-section is a ring with area:
$$ \pi(R^2 - r^2) $$
where $R$ is outer radius and $r$ is inner radius.
Important Points#
- Radius varies with $x$, so express $r$, $R$ in terms of $x$.
- Integrate cross-sectional area over the interval to find the volume.
- Examples include revolving $y = 3\sqrt{x}$ from $x=1$ to $x=4$:
$$ V = \pi \int_1^4 (3\sqrt{x})^2 , dx = 9\pi \int_1^4 x , dx $$
which simplifies accordingly.
Geometric Interpretation#
- This method generalizes classical volumes of cylinders, cones, spheres by accumulating many thin circular slices.
- Sides must be perpendicular to the axis of revolution to apply the method.
Summary#
Professor Leonard’s lecture details the disks and washers methods for volumes of revolution: approximate volume by summing circular cross-sections’ areas times thickness, then integrate to get exact volume. This approach extends the integral concept from areas to volumes, with radius functions determined by curves being revolved, and constants factored for simpler integration.