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Volumes by Disks, Washers, and Shells cheat sheet - grade 11-12

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Calculus Grade 11-12

Volumes by Disks, Washers, and Shells Cheat Sheet

A printable reference covering disk, washer, and shell volume formulas, radius setup, axis choice, and integration bounds for grades 11-12.

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Volumes by disks, washers, and shells help students find the volume of a solid formed by rotating a region around an axis. This cheat sheet explains when to use each method and how to build the correct integral. Students need it because most errors come from choosing the wrong radius, bounds, or variable.

The goal is to turn a picture of a region into a clear volume formula.

Key Facts

  • The disk method is used when cross sections perpendicular to the axis of rotation are solid circles, with volume V=π∫ab[R(x)]2 dxV = \pi \int_a^b [R(x)]^2\,dx or V=π∫cd[R(y)]2 dyV = \pi \int_c^d [R(y)]^2\,dy.
  • The washer method is used when cross sections perpendicular to the axis of rotation have a hole, with volume V=π∫ab([R(x)]2−[r(x)]2) dxV = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right)\,dx.
  • For washers, the outer radius RR is the distance from the axis of rotation to the farther curve, and the inner radius rr is the distance to the closer curve.
  • The shell method is used with slices parallel to the axis of rotation, with volume V=2π∫ab(radius)(height) dxV = 2\pi \int_a^b (\text{radius})(\text{height})\,dx or V=2π∫cd(radius)(height) dyV = 2\pi \int_c^d (\text{radius})(\text{height})\,dy.
  • A shell radius is the distance from the slice to the axis of rotation, such as r(x)=xr(x) = x for rotation around the yy-axis or r(x)=x−ar(x) = x - a for rotation around x=ax = a.
  • A shell height is the length of the slice through the region, often h(x)=f(x)−g(x)h(x) = f(x) - g(x) for vertical slices or h(y)=xright−xlefth(y) = x_{\text{right}} - x_{\text{left}} for horizontal slices.
  • Use dxdx when slices are vertical and bounds are xx-values, and use dydy when slices are horizontal and bounds are yy-values.
  • If the axis of rotation is not one of the coordinate axes, radii must be written as distances, such as R(x)=5−f(x)R(x) = 5 - f(x) or r(y)=g(y)−2r(y) = g(y) - 2.

Vocabulary

Disk Method
A volume method that uses circular cross sections with no hole, usually written as V=π∫[R]2 dxV = \pi \int [R]^2\,d x or V=π∫[R]2 dyV = \pi \int [R]^2\,d y.
Washer Method
A volume method that uses cross sections shaped like washers, with volume found by subtracting the inner circular area from the outer circular area.
Shell Method
A volume method that adds thin cylindrical shells using V=2π∫(radius)(height) dxV = 2\pi \int (\text{radius})(\text{height})\,d x or V=2π∫(radius)(height) dyV = 2\pi \int (\text{radius})(\text{height})\,d y.
Axis of Rotation
The line around which a two-dimensional region is rotated to create a three-dimensional solid.
Outer Radius
In the washer method, the outer radius is the greater distance from the axis of rotation to the boundary of the region.
Inner Radius
In the washer method, the inner radius is the smaller distance from the axis of rotation to the boundary of the hole.

Common Mistakes to Avoid

  • Using R−rR - r instead of R2−r2R^2 - r^2 for washers is wrong because washer area comes from subtracting circle areas, so the correct expression is π(R2−r2)\pi(R^2 - r^2).
  • Forgetting that radii are distances is wrong because a radius cannot be negative, so expressions like R(x)R(x) should measure positive distance from the axis of rotation.
  • Mixing dxdx bounds with yy-based radii is wrong because the variable of integration must match the slice direction and the interval limits.
  • Choosing shells when the height is not written as top minus bottom or right minus left causes errors because shell height must represent the full length of the slice inside the region.
  • Ignoring an axis shift such as x=3x = 3 or y=−2y = -2 is wrong because radii must be measured from that shifted line, not automatically from the xx-axis or yy-axis.

Practice Questions

  1. 1 Find the volume when the region under y=x2y = x^2 from x=0x = 0 to x=2x = 2 is rotated about the xx-axis using disks.
  2. 2 Set up and evaluate the washer integral for the region between y=4y = 4 and y=x2y = x^2 rotated about the xx-axis.
  3. 3 Use cylindrical shells to find the volume when the region under y=6−xy = 6 - x from x=0x = 0 to x=6x = 6 is rotated about the yy-axis.
  4. 4 A region is easier to describe with vertical slices, but it is rotated around a vertical axis. Explain why the shell method may be simpler than washers.

Understanding Volumes by Disks, Washers, and Shells

The most reliable first step is to draw one representative slice before writing any integral. Mark the axis of rotation clearly. Then draw the slice in the original flat region and imagine it turning once.

A slice that touches the axis makes a filled circle. A slice that stops before the axis leaves an empty center.

This picture tells you whether a cross section has material all the way through. It prevents a common mistake where students subtract an inner radius even though there is no hole, or forget to subtract one when a hole is present.

Slice direction controls more than the differential. For disk and washer work, slices must cross the axis at a right angle. For shell work, slices run in the same direction as the axis.

This rule is useful when a graph is given with functions of x but a horizontal slice would require solving for x in terms of y. Sometimes that algebra is easy, and sometimes it creates two branches or awkward bounds.

In those cases, shells can be the cleaner method. The best method is usually the one that describes the region with one simple slice length and one clear distance to the axis.

Distances deserve extra care because a radius can never be negative. If the axis lies at a fixed horizontal or vertical value, find each radius by measuring the gap between the curve and that line. Do not rely only on the order in which expressions are written.

For example, when the axis is above a region, the lower curve may be farther from the axis than the upper curve. A quick sketch with a short distance segment often reveals which expression must be the outer radius.

The same idea applies to shells. Their height is the part of the slice inside the region, not the height of a function measured from an axis unless that axis is actually a boundary of the region.

Many volume problems change character at an intersection, a turning point, or a place where the axis passes through the region. One formula may not work across the whole interval. Split the volume into separate integrals when the top and bottom curves switch, the left and right curves switch, or the outer radius changes.

This is not extra work for its own sake. It reflects that the solid has different cross sections in different places. Finding intersection values before setting bounds gives the integration limits a geometric meaning instead of making them numbers chosen from a graph by guesswork.

Volumes have cubic units, so units provide a useful final check. If every graph distance is measured in centimeters, the answer is in cubic centimeters. A volume must be nonnegative.

If an integrand becomes negative, the radius order, shell height, or bounds need attention. You can estimate reasonableness by comparing the solid with a cylinder or box that contains it.

These methods appear in manufacturing when a turned object is shaped on a lathe, in engineering when pipes have hollow interiors, and in design when curved containers need a capacity estimate. The calculus works because integration adds many very thin pieces whose individual volumes are easy to describe.