AP Unit 8 Problems
Thirty problems for AP Unit 8. Try the problem before opening its separate solution. Write the mathematical setup and the reason your method applies.
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Problem 1
Find the average value of f(x)=x² on [0,3] and a point where f takes that value.
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Problem 2
| t (min) | 0 | 2 | 5 |
|---|---|---|---|
| T(t) (°C) | 10 | 14 | 20 |
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Problem 3
A particle has v(t)=2t−2 m/s on [0,3] and s(0)=5 m. Find its final position and total distance.
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Problem 4
Acceleration is a(t)=6t m/s² and v(0)=−2 m/s. Find v(2) and the first nonnegative time the particle changes direction.
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Problem 5
A velocity graph consists of a triangle above the axis on [0,2] with height 4 m/s and a triangle below on [2,5] with depth 2 m/s. Find displacement and distance.
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Problem 6
A tank starts with 10 L and has net rate 6−2t L/min for 0≤t≤4. Find its maximum volume and the time it occurs.
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Problem 7
Inventory starts at 50 items. Inflow is 5t items/hour and outflow is 2 items/hour on [0,3]. Find the inventory at hour 3.
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Problem 8
A rate r(t) is measured in dollars/day. Explain the meaning and units of ∫₂⁷ r(t)dt, and how to use it if the account balance at day 2 is $100.
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Problem 9
Find the area enclosed by y=x and y=x².
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Problem 10
Find the area enclosed by y=4 and y=x².
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Problem 11
Find the area between x=y and x=y² using horizontal strips.
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Problem 12
Find the area of the region 0≤y≤2, 0≤x≤y².
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Problem 13
Find the total area between y=x and y=x³ on [−1,1].
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Problem 14
Find the area between y=sin x and the x-axis on [0,2π]. Also give the signed integral.
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Problem 15
The base lies between y=x and y=x² on [0,1]. Cross sections perpendicular to the x-axis are squares. Find the volume.
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Problem 16
On 0≤x≤2, each rectangular cross section has base 2−x and height three times its base. Find the volume.
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Problem 17
The base lies between y=x and y=x² on [0,1]. Perpendicular-to-x slices are equilateral triangles with this vertical segment as a side. Find the volume.
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Problem 18
The base lies between y=0 and y=2−x on [0,2]. Perpendicular-to-x sections are semicircles with diameter 2−x. Find the volume.
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Problem 19
Rotate the region 0≤y≤√x, 0≤x≤4, about the x-axis. Find the volume.
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Problem 20
Rotate the region 0≤x≤√y, 0≤y≤4, about the y-axis. Find the volume.
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Problem 21
Rotate the region 1≤y≤1+x, 0≤x≤2, about y=1. Find the volume.
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Problem 22
Rotate the region x≤y≤2, 0≤x≤2, about the x-axis. Find the volume.
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Problem 23
Rotate the region x≤y≤2, 0≤x≤2, about y=3. Find the volume.
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Problem 24
Use a calculator to find the area between y=e^(−x²) and y=x on [0,1], to three decimal places. Show the intersection and the integral setup.
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Problem 25
Find the length of y=3x+1 on [0,2]. Verify it using endpoint distance.
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Problem 26
Find the arc length of y=(2/3)x^(3/2) on [0,3].
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Problem 27
Use a calculator to find the length of y=x²/2 on [0,2], to three decimals.
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Problem 28
A curve is given by x=y²/2 for 0≤y≤2. Write a length integral and approximate it to three decimals with a calculator.
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Problem 29
For y=x on [0,1], compare horizontal distance, area under the graph, and arc length. Give units for each.
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Problem 30
The graph y=|x| on [−1,1] has a corner at 0. Find its length and explain why you split the interval.
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More chapter problems
Try these problems before opening the solutions. Show your setup, the hypotheses behind your method, and what your answer means.
Build a triangular solid from a curved base
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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →
A diameter makes a semicircular volume
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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →
Rotate one actual base about two axes
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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →
Does the rotating segment actually leave a hole?
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Worked solution and rubric →Revisit the explanation →Read the relevant lesson →