AP Calculus AB: Volumes with Known Cross Sections — Drill 1

About This Drill

AP Calculus AB: Volumes with Known Cross Sections — Drill 1 is a Math practice drill covering Volumes with Known Cross Sections. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice computing volumes of solids with known cross sections perpendicular to the x-axis, including squares, semicircles, and equilateral triangles. These AP Calculus AB problems require correctly identifying the side or diameter length from the bounding curves and applying the appropriate area formula.

Questions in This Drill

1. The region in the xy-plane is bounded above by \( y = 2x \) and below by the x-axis on \( [0, 3] \). Cross sections of the solid perpendicular to the x-axis are squares with side length equal to the height of the base. Which integral gives the volume of the solid?
2. The base of a solid is the region bounded by \( y = \sqrt{4 - x^2} \) and the x-axis. Cross sections perpendicular to the x-axis are semicircles with diameter equal to the height of the base. Which integral gives the volume of the solid?
3. The base of a solid is the region bounded by \( y = 4 - x^2 \) and \( y = 0 \). Cross sections perpendicular to the x-axis are equilateral triangles with one side lying in the base. Which integral gives the volume of the solid?
4. The table below shows selected values of \( h(x) \), the vertical distance between the upper and lower boundaries of a region. Cross sections of the solid perpendicular to the x-axis are squares with side length \( h(x) \).
   
   

   x	0	1	2	3
   h(x)	1	3	2	4

   
   Using a left Riemann sum with three equal subintervals on \( [0, 3] \), approximate the volume of the solid.
5. A solid has a base in the xy-plane bounded by \( y = \cos x \) and \( y = 0 \) on \( \left[0, \dfrac{\pi}{2}\right] \). Three students each compute a volume using this base but with different cross sections perpendicular to the x-axis: Student 1 uses squares, Student 2 uses equilateral triangles, and Student 3 uses semicircles with diameter equal to the base height. Which of the following correctly ranks the three volumes from greatest to least?