Code Scholars Tutoring · AP Calculus BC
13 topics across 5 lessons, with explanations, worked examples, and self-checks that develop complete mathematical reasoning.
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.
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.
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.
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.
Topics 8.13 · BC only
Arc length accumulates small distances along a curve. For a graph , a short horizontal change and its corresponding vertical change form the legs of a small right triangle. The resulting length element explains the square root and the added in the graph formula.
