Practice a College Board-style free response question on area between curves and volumes of revolution. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 8 · Applications of Integration
Let R be the region in the first quadrant bounded above by y = √x and below by y = x/2.
A
Find the x-coordinates where the two curves intersect, and set up (do not yet evaluate) the integral for the area of region R.
✓ Model answer
Set √x = x/2. Squaring: x = x²/4, so x² − 4x = 0, x(x − 4) = 0, giving x = 0 and x = 4. On (0, 4), √x ≥ x/2, so the area is A = ∫ from 0 to 4 of (√x − x/2) dx.
Why it scores: Correct intersection points x = 0, 4 and a correct top − bottom integrand with proper limits.
B
Evaluate the integral from part A to find the area of region R.
✓ Model answer
A = ∫₀⁴ (x^(1/2) − x/2) dx = [(2/3)x^(3/2) − x²/4] from 0 to 4 = ((2/3)(8) − 16/4) − 0 = 16/3 − 4 = 4/3.
Why it scores: Correct antiderivative (2/3)x^(3/2) − x²/4, correctly evaluated to 4/3. An antidifferentiation or arithmetic slip loses the point.
C
Region R is revolved about the x-axis. Set up (do not evaluate) the integral for the volume of the resulting solid.
✓ Model answer
Revolving about the x-axis leaves a gap between the curves and the axis, so use the washer method with outer radius R = √x and inner radius r = x/2: V = π∫ from 0 to 4 of ((√x)² − (x/2)²) dx = π∫₀⁴ (x − x²/4) dx.
Why it scores: Chooses the washer method with correct outer/inner radii and squares each — not (R − r)². Using a disc (ignoring the hole) would lose credit.
How to score points on AP Calculus FRQs
Find intersection points first — they become the limits of integration.
Integrate top − bottom (or right − left) so the slice height is positive.
Choose disc vs. washer by whether a gap to the axis creates a hole.
Square each radius separately in a washer — never (R − r)².
Answer the verb. "Set up" wants the integral; "evaluate" wants the number.