Calculus AB — area between curves and volumes with known cross sections

Area and volume set-ups appear on almost every AP Calculus AB exam, in multiple-choice form and as parts of a free-response question about a region drawn between two curves. The calculus in them is short; the points are lost in the set-up: the wrong limits, the wrong curve on top, a radius confused with a diameter, a slice taken in the wrong direction. This material trains those set-ups, following topics 8.4 to 8.8 of the course framework.

The twelve quiz questions start with area. You find the area between y = x and y = x² after locating where they meet, the area between y = x³ and y = x on an interval where the two curves swap places, so the integral has to be split rather than taken in one piece, and the area between x = y² and x = y + 2, where horizontal slices and a right-minus-left integrand make the work much shorter. One question asks only which integral is the correct set-up for the region between sine and cosine, and two more ask for the area under a square-root curve and inside a parabola cut off by a horizontal line.

The second half builds solids on those same regions. The cross sections perpendicular to the x-axis are squares, rectangles of fixed height, semicircles whose diameter lies in the base, equilateral triangles, and isosceles right triangles with a leg in the base; a last solid is sliced perpendicular to the y-axis, so its side length has to be written in terms of y. Each time the volume is the integral of the cross-sectional area, and the only real question is what that area is at position x or at height y.

The wrong options come from the mistakes these problems actually produce: a signed integral that cancels to zero where an area was wanted, a diameter used as a radius, a missing factor of one half or of the square root of three over four, half the width taken as the side, a slice perpendicular to the wrong axis. The explanation after each answer shows the set-up line by line and names the mistake behind the most tempting wrong value.

The flashcards collect the formulas: area as top minus bottom or right minus left, volume as the integral of A(x), and the area of each standard cross section, including the isosceles right triangle with its hypotenuse in the base.

This is independent practice written by Zestly, based on the published AP Calculus AB course framework; it is not produced or endorsed by the College Board.

  • Find the limits of integration from the intersection points of two curves
  • Compute the area between curves with vertical slices, splitting the integral where the curves cross
  • Compute an area with horizontal slices as right minus left
  • Choose the correct integral set-up for a region
  • Find volumes of solids with square, rectangular, semicircular, equilateral-triangle and isosceles-right-triangle cross sections
  • Set up a volume with cross sections perpendicular to the y-axis

Practice material written by Zestly, based on the College Board AP Calculus AB course framework (Unit 8: Applications of Integration, topics 8.4–8.8: area between curves with respect to x and y, curves that intersect at more than two points, volumes with square, rectangular, triangular and semicircular cross sections), 2026–27 course and exam description.

Sample question

Which integral correctly represents the area between $y = \sin(x)$ and $y = \cos(x)$ from $x = 0$ to $x = \pi/4$?

See the answer

$\int_0^{\pi/4} (\cos(x) - \sin(x)) dx$

On $[0, \pi/4]$, $\cos(x) \ge \sin(x)$, so the area is the integral of the top function minus the bottom function, which is $\cos(x) - \sin(x)$.

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