Unit 5 · Core unit
The graded skill is the setup, not the antiderivative. Almost every lost mark here comes from the diagram, the slice, or the choice of variable.
What a strong answer looks like
A strong answer shows the representative slice before the integral: its dimensions, which curve is which, and which variable it is measured in.
Chapter introduction
The graded skill is the setup, not the antiderivative. Almost every lost mark here comes from the diagram, the slice, or the choice of variable.
In Calculus Honors, success means moving among words, symbols, tables, and graphs while preserving the meaning of each quantity. Use the lessons below as connected notes: explain the concept, carry out the procedure, test the result, and interpret it in context.
Detailed study notes
Lesson 1
The integrand is the vertical distance between the curves, upper minus lower, and the interval runs between their intersections.
Sketch first, find the intersections, and split the integral wherever the curves swap order.
Common mistake
Subtracting in the wrong order, or integrating across a crossing point without splitting.
Guided study move
Find an area where the curves cross inside the interval, so the integral must be split.
Lesson 2
Disks and washers stack cross-sections perpendicular to the axis; shells wrap cylinders around it. The washer subtracts squares of radii, not the square of a difference.
Draw the slice, decide whether it is perpendicular or parallel to the axis, and write its area or volume before integrating.
Common mistake
Squaring the gap between two curves instead of subtracting their squares.
Guided study move
Set up the same solid by both washers and shells and confirm the two integrals agree.
Lesson 3
The average value divides accumulated total by interval width. Integrating a rate gives net change, and integrating its absolute value gives total amount.
Check units at every step: a rate times a duration must leave the quantity being accumulated.
Common mistake
Reporting displacement when total distance was asked for.
Guided study move
Compute displacement and total distance for the same velocity function and explain the difference.
Formula and visual reference
Chapter synthesis
Modeling lab
A region bounded by two curves is revolved about a line that is not an axis. Set up the integral, stating each radius as a distance from the line.
Modeling drill
A region bounded by two curves is revolved about a line that is not an axis. Set up the integral, stating each radius as a distance from the line.
