volume with regualr cross-section

One of the major applications of integration is to find the volumes of various solid figures.

Volume of Solids with Regular Cross-sections This is where to start with volume problems. After all, solids of revolution are just a special case of solids with regular cross-sections.

Volumes of Revolution

Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

Visualizing Solid Figures

Visualizing Solid Figures 1 Making models et al.
Visualizing Solid Figures 2 Using Winplot to visualize solids with regular cross-sections.
Visualizing Solid Figures 3 Using Winplot to visualize the Disk and Washer methods.
Visu alizing Solid Figures 4 Using Winplot to visualize finding volumes by cylindrical shells. While no problem on the AP Calculus exams will require a solution by the Cylindrical Shell method, students may use the method and be eligible for full credit. So this is “enrichment.”
Challenge – the half-full problem comparing the disk and shell methods and Visualizing Solid Figures 5 the detailed solution

Why you Never Need Cylindrical Shells

Painting a Point

Revised and update October 22, 2018

AP Type Questions 4

Given equations that define a region in the plane students are asked to find its area and the volume of the solid formed when the region is revolved around a line or used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since 1969 on the AB exam and all but one year on the BC exam.

If this appears on the calculator active section, it is expected that the definite integrals will be evaluated on a calculator. Students should write the definite integral with limits on their paper and put its value after it. It is not required to give the antiderivative and if students give an incorrect antiderivative they will lose credit even if the final answer is (somehow) correct.

There is a calculator program available that will give the set-up and not just the answer so recently this question has been on the no calculator allowed section. (The good news is that the integrals will be easy or they will be set-up but do not integrate questions.)

What students should be able to do:

Find the intersection(s) of the graphs and use them as limits of integration (calculator equation solving). Write the equation followed by the solution; showing work is not required. Usually no credit is earned until the solution is used in context (as a limit of integration).
Find the area of the region between the graph and the x-axis or between two graphs.
Find the volume when the region is revolved around a line, not necessarily an axis, by the disk/washer method. (Shell method is never necessary, but is eligible for full credit if properly used).
Find the volume of a solid with regular cross-sections whose base is the region between the curves. But see 2009 AB 4(b)
Find the equation of a vertical line that divides the region in half (area or volume). This involves setting up and solving an integral equation where the limit is the variable for which the equation is solved.
For BC only – find the area of a region bounded by polar curves:

$\displaystyle A=\tfrac{1}{2}{{\int_{{{t}_{1}}}^{{{t}_{2}}}{\left( r\left( t \right) \right)}}^{2}}dt$

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

For some previous posts on this subject see January 9, 11, 2013

Teaching Calculus

The pleasure lies not in discovering the truth, but in searching for it.

Tag Archives: volume with regualr cross-section

Applications of Integration – Volume

Area and Volume Questions

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