Applications of Integration: AP Calculus AB Math Test – PDF Download

AP Calculus Applications Integration Test: questions 1–5

Question 1: Average value (2 points)

Let \(f(x)=x^{3}\) on \([0,2]\).

1. Find the average value of \(f\). (1 point)
2. Find the value of \(c\) in \((0,2)\) where \(f(c)\) equals that average. Round to three decimal places. (1 point)

Question 2: Particle motion (3 points)

A particle moves on a line with velocity \(v(t)=t^{2}-6t+8\) for \(0\le t\le 5\), and \(s(0)=1\).

1. Find the position \(s(5)\). (1 point)
2. Find the total distance traveled from \(t=0\) to \(t=5\). (2 points)

Question 3: Area between curves (2 points)

Find the area of the region bounded by \(y=6x-x^{2}\) and \(y=x^{2}\), shown above.

Question 4: Area using y (2 points)

Find the area of the region bounded by \(x=y^{2}\) and \(x=6-y\).

Question 5: Semicircle cross sections (3 points)

The base of a solid is the region between \(y=x^{2}\) and \(y=4\). Cross sections perpendicular to the \(x\)-axis are semicircles whose diameters lie in the base. Find the volume.

AP Calculus Applications Integration Test: questions 6–8

Question 6: Washer method (3 points)

The region bounded by \(y=\sqrt{x}\) and \(y=\frac{x}{2}\) is revolved about the \(x\)-axis. Find the volume of the solid.

Question 7: Rates of change (3 points)

The number of people \(A(t)\) in a stadium, in hundreds, changes at the rate \((2+t)-(8-t)\), where \(2+t\) is the rate in and \(8-t\) is the rate out, in hundreds per hour, for \(0\le t\le 8\). At \(t=0\), \(A=50\).

1. Find \(A(6)\). (1 point)
2. At what time is \(A\) least? Justify. (1 point)
3. Find the least value of \(A\). (1 point)

Question 8: Area with three intersections (2 points)

Find the total area of the regions enclosed by \(y=x^{3}-9x\) and the \(x\)-axis.

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AP Calculus AB Applications of Integration test

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