Code Scholars Tutoring · AP Calculus AB
10–15% of the multiple-choice section. Study the concepts, conditions, and reasoning for all 12 topic codes in this unit.
Topics 8.1, 8.2, 8.3
The average value of a function is its integral divided by the interval length; it is different from average rate of change. Integrating velocity gives displacement, while integrating speed gives distance. To recover a quantity from its rate, add the accumulated change to its initial amount. Inflow minus outflow gives net rate, not total throughput. Track units at every stage.
Read lesson and worked example →Topics 8.4, 8.5, 8.6
Find all intersections that bound the region and identify which curve is above or to the right on each subinterval. Vertical slices integrate top minus bottom with respect to ; horizontal slices integrate right minus left with respect to . Split where the ordering changes. Geometric area is nonnegative, so an unsplit signed integral may not answer the question.
Read lesson and worked example →Topics 8.7, 8.8
A solid volume is the integral of its cross-sectional area. First identify the base length from the region, then convert that length to the area of the stated shape. For a semicircle whose diameter is the base width , area is . For an equilateral triangle it is . Rectangular sections require the given height relationship.
Read lesson and worked example →Topics 8.9, 8.10, 8.11, 8.12
For rotation, radius is distance from the rotation axis. A disk has one radius; a washer removes an inner disk from an outer disk. Both radii must be measured in the same slicing direction. A shifted axis changes the radius expressions, not the general formula. Choose vertical slices for rotation about a horizontal axis and horizontal slices for rotation about a vertical axis when using disks or washers.
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