This ap calculus applications integration worksheet gives 24 problems on applications of integration, ordered from easy to hard, with a separate answer key.
These 24 problems practice AP Calculus AB Unit 8: average value, particle motion, area between curves, volumes with cross sections, discs and washers, and accumulation with rates. They move from one-step integrals to multi-part word problems.
Sketch each region, label limits, and write the integral before evaluating it.
AP Calculus Applications Integration Worksheet: problems 1–5
Problem 1: Average value of a polynomial
Determine the average value of \(g(x)=3x^{2}-2x\) over the interval \([1,4]\).
Problem 2: Average value of sine
Find the average value of \(f(x)=\sin x\) on \([0,\pi]\). Give an exact answer and a decimal rounded to four places.
Problem 3: Average value of a quadratic
Find the average value of \(f(x)=3x^{2}-2x+1\) on \([1,3]\).
Problem 4: Where the average value occurs
For \(f(x)=x^{2}\) on \([0,3]\), find the value of \(c\) in \((0,3)\) at which \(f(c)\) equals the average value.
Problem 5: Displacement and distance
A particle moves along a line with velocity \(v(t)=t^{2}-4t+3\) for \(0\le t\le 4\). Find the displacement and the total distance traveled.
AP Calculus Applications Integration Worksheet: problems 6–10
Problem 6: Final position
A particle has velocity \(v(t)=3t^{2}-6t\) and position \(s(0)=2\). Find \(s(2)\).
Problem 7: Cosine motion
A particle has velocity \(v(t)=\cos t\) for \(0\le t\le\pi\). Find the displacement and the total distance.
Problem 8: Area between a parabola and a line
Find the area of the region enclosed by the line \(y=3x+4\) and the parabola \(y=x^{2}\).
Problem 9: Area with two intersection points
Find the area of the region bounded by \(y=x+2\) and \(y=x^{2}\), shown above.
Problem 10: Area between exponentials
Find the area between \(y=e^{x}\) and \(y=e^{-x}\) for \(0\le x\le\ln 3\).
AP Calculus Applications Integration Worksheet: problems 11–15
Problem 11: Area using y
Find the area of the region bounded by \(x=y^{2}\) and \(x=2y\) by integrating with respect to \(y\).
Problem 12: Three intersection points
Find the total area enclosed by \(y=x^{3}\) and \(y=x\).
Problem 13: Sine and cosine area
Find the area enclosed by \(y=\sin x\) and \(y=\cos x\) between consecutive intersections at \(x=\frac{\pi}{4}\) and \(x=\frac{5\pi}{4}\).
Problem 14: Square cross sections
The base of a solid is the region between \(y=\sqrt{x}\) and \(y=\frac{x}{2}\). Cross sections perpendicular to the \(x\)-axis are squares. Find the volume.
Problem 15: Semicircle cross sections
The base of a solid is the region bounded by \(y=\sqrt{x}\), the \(x\)-axis and \(x=4\). Cross sections perpendicular to the \(x\)-axis are semicircles whose diameters lie in the base. Find the volume.
AP Calculus Applications Integration Worksheet: problems 16–20
Problem 16: Triangle cross sections
The base of a solid is the region between \(y=6x-x^{2}\) and the \(x\)-axis. Cross sections perpendicular to the \(x\)-axis are equilateral triangles with a side in the base. Find the volume.
Problem 17: Disc method about the x-axis
The region bounded by \(y=\sqrt{x}\), the \(x\)-axis and \(x=4\) is revolved about the \(x\)-axis. Find the volume.
Problem 18: Disc method about the y-axis
The region bounded by \(y=x^{2}\), the \(y\)-axis and \(y=4\) is revolved about the \(y\)-axis. Find the volume.
Problem 19: Washer about the x-axis
The region between \(y=x\) and \(y=x^{3}\) in the first quadrant is revolved about the \(x\)-axis. Find the volume.
Problem 20: Washer about a horizontal line
The region between \(y=x\) and \(y=x^{2}\) is revolved about the line \(y=-1\). Find the volume.
AP Calculus Applications Integration Worksheet: problems 21–24
Problem 21: Rotation about a vertical line
The region bounded by \(y=\sqrt{x}\), the \(x\)-axis and \(x=4\) is revolved about the line \(x=4\). Find the volume.
Problem 22: Water in a tank
A tank holds 100 gallons at \(t=0\). Water flows in at \(5+2t\) gallons per minute and out at 9 gallons per minute for \(0\le t\le 6\). Find the amount at \(t=6\), and find when the amount is least.
Problem 23: Oil leak
Oil leaks from a tank at a rate of \(12e^{-0.2t}\) gallons per hour. How many gallons leak during the first 5 hours? Round to the nearest hundredth.
Problem 24: Average temperature
The temperature in a greenhouse is \(T(t)=60+10\sin(\frac{\pi t}{12})\) degrees Fahrenheit, \(0\le t\le 12\), where \(t\) is in hours after 6 a.m. Find the average temperature on this interval, to the nearest tenth of a degree.
How to use this ap calculus applications integration worksheet
Print this ap calculus applications integration worksheet as a PDF, solve every problem on paper, then compare with the answer key.
See every chapter on the AP Calculus AB math page.