Unit 8 - Day 15

Writing an AP Calc Assessment 
  • Include multiple choice and free response items

  • Write questions that reflect learning targets and success criteria

  • Determine scoring rubrics for FRQs before administering the assessment

Questions to Include
  • Calculate volumes of revolution using disk cross sections and definite integrals

  • Calculate volumes of revolution using washer cross sections and definite integrals

  • Calculate volumes of revolution using known cross sections and definite integrals

  • Set up and/or evaluate expressions for the volume of a region revolved around axes other than the 
    x-axis or y-axis

  • Bounding functions requiring calculator assistance to define points of intersection

Grading Tips

Complex integrals are required to accurately represent the volumes generated in Topics 8.7 through 8.12.  To best assess student understanding, consider awarding points for specific components in a student’s response.

Simply using the correct orientation of slices or washers (dx vs. dy) and the corresponding limits of integration may be the focus for a particular question on your quiz and may constitute all the points for that problem. For volumes incorporating washers or other cross section regions, emphasis should be placed on correct formatting of the integrand. 

When asking students to evaluate their definite integrals, omitting constants (pi, incorrect denominators, etc.) typically results in a loss of point only for the final numeric answer. Students should not lose points for an incorrect intermediate expression AND the final value if their only mistake is a misplaced constant. 

Encourage students to sketch a representative washer or slice when needed; just don’t assess the artwork!


Students performed best when asked to find volumes by solid slices: especially when the axis of rotation was a major axis. These shapes are easy to visualize and the integrands are manageable.

As expected, washer problems presented more challenges than slices, especially when a region was revolved around a distant axis.  Our students were successful horizontal slices and washers and also were able to find the correct limits of integration along the y-axis. 

Thanks to consistent practice throughout the year, students were comfortable finding points of intersection and storing those values in their calculator for easy access later. The time saved during a test is well worth the time spent teaching this skill!

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