Assignments App of Int - Norwood Norfolk School
AP Calculus Assignments: Areas and Volumes
|Day |Topic |Assignment |
|1 |Areas between curves I |HW Areas and Volumes - 1 |
|2 |Areas between curves II |HW Areas and Volumes - 2 |
|3 |Practice **QUIZ** |HW Areas and Volumes - 3 |
|4 |Volume: Disk method |HW Areas and Volumes - 4 |
|5 |Volume: Washer method |HW Areas and Volumes - 5 |
|6 |Practice **QUIZ** |HW Areas and Volumes - 6 |
|7 |Volumes of solids with known cross sections |HW Areas and Volumes - 7 |
|8 |Review |HW Areas and Volumes - Review |
|9 |**TEST** | |
AP Calculus Areas and Volumes - 1
Draw an appropriate diagram and find the area bounded by the given curves without using your calculator.
1. [pic]and [pic] 2. [pic] and [pic]
Draw an appropriate diagram and find the area bounded by the given curves. You must show the integral setup, including limits and the equation(s) from which the limits came.
3. [pic], [pic] and [pic] 4. [pic] and [pic]
5. [pic] and [pic] 6. y = x2, y = 8 – x2 and the x-axis
7. Find the area of one of the smaller of the regions bounded by the graphs of y = cos x, y = k, 0 ( k ( 1.
AP Calculus Areas and Volumes - 2
Find the area of the shaded region.
1. 2.
3. Find the area between the graphs of y2 = 2x + 6 and y = x – 1
a. Using rectangles oriented vertically.
b. Using rectangles oriented horizontally.
4. Use horizontal rectangles to find the area in the first quadrant bounded by the graphs of [pic]
and y = ln x.
5. Find the area bounded by the graphs of[pic] and [pic].
6. Find the area bounded by the graphs of [pic] and [pic].
AP Calculus Areas and Volumes - 3
1. Find the area shown at right enclosed by the curves [pic] and y = 2. You should be able to do this without using your calculator.
2. Find the area of the region bounded by the parabola [pic], the tangent line to this parabola at (1, 1) and the x-axis.
3. Find the number b such that the line y = b divides the region bounded by the curves [pic] and y = 4 into two equal parts.
4. a. For what values of m will the graphs of y = mx and [pic] enclose an area?
b. Find, in terms of m, the first-quadrant area enclosed by the graphs of y = mx and [pic].
5. a. Find, in terms of k, the area of the region bounded by the graphs of y = xk, [pic], k > 0, and the y-axis. Note: the graphs intersect at the point (1, 1).
b. Find the value of k that will minimize the area from part (a).
6. a. Write an integral expression for the area under the graph of y = xe1 – x on the interval 0 ( x ( k.
b. Find the rate of change of the area from part (a) with respect to k.
AP Calculus Areas and Volumes - 4
1. Find the volume of the solid formed by rotating the region bounded by the graphs of [pic], x = 1 and
y = 0 about the x-axis.
2. Find the volume of the solid formed by rotating the region bounded by the graphs of [pic], y = 0, x = 0 and x = ln 3 about the x-axis.
3. Find the volume of the solid formed by rotating the region bounded by the graphs of [pic], y = 4 and x = 0 about the y-axis.
4. Verify using an integral that the volume of a sphere of radius r is given by V = πr3. (Hint: rotate part of the circle x2 + y2 = r2 around the x-axis.)
5. A right circular cone of height h and radius r is generated by rotating the line segment joining (0, h) to
(r, 0) about the y-axis. Verify using an integral that the volume of this cone is given by V = πr2h.
6. This problem is actually rather easy. It is included here partly for its enrichment value but mostly because I enjoy driving my students nuts.*
a. Set up the appropriate integral to find the area under the graph of [pic] to the right of the line x = 1.
b. This is one form of an improper integral (a topic not required in A.P. Calc AB). The correct way to evaluate it is to change the upper limit to b, do the integral, then take the limit as b ( (. Do it.
c. Now find the volume that results if the area under the graph above is rotated about the x-axis. This solid is called Gabriel’s Horn.
d. To find the surface area of the solid from part c (another topic not included in A. P Calc AB), we need to evaluate [pic]. Note that [pic] whenever x ( 1. Use that fact and your common sense (along with a peek back at part a above) to find the surface area of Gabriel’s Horn.
* Grammar note: Microsoft Word did not like this sentence. It suggested that the word “students” should be possessive.
AP Calculus Areas and Volumes - 5
1. Find the volume of the solid obtained by rotating the region bounded by the graphs of [pic] and [pic] about the x-axis.
2. Find the volume of the solid obtained by rotating the first-quadrant region bounded by the graphs of [pic], [pic] and the y-axis about the x-axis.
3. Find the volume of the solid obtained by rotating the region bounded by the graphs of [pic], [pic] and [pic] about the y-axis.
(This assignment continued on the next page.)
4. Find the volume of the solid obtained by rotating the region bounded by the graphs of [pic] and [pic] about the line y = 2.
5. The region in the first quadrant enclosed by the curves x = 4y and[pic] is rotated about the line x = 8. Find the volume of the resulting solid.
6. Find the volume of the solid obtained by rotating the region in the previous problem about the line y = 2.
AP Calculus Areas and Volumes - 6
1. The cooling tower for a nuclear power plant is 120 meters high and has circular cross-sections. At height h, the radius of the tower is given by [pic], where
0 ( h ( 120. The units of r and h are meters.
a. Find the volume of the cooling tower.
b. Find the average value of the radius of the tower.
2. a. Find the volume that results from rotating the region between the graph of [pic]and the
x-axis around the x-axis.
b. Find the value of c so that the vertical plane x = c will divide the volume from part a into two equal
parts.
3. A circular paraboloid results when a segment of a parabola (which includes the vertex) is rotated about its axis.
a. If the origin is at the vertex of the paraboloid, then equation of the generating parabola is y = ax2. Find the value of a if the paraboloid has a radius of r and a height of h.
b Find the volume of the paraboloid in terms of r and h.
4. A spherical cap is that part of a sphere cut off by a plane. Set up an integral to find the volume of a spherical cap of height h cut off from a sphere of radius r. It is not necessary to evaluate your integral. Note: a is the radius of the base of the cap. The volume may be expressed in terms of h and r or in terms of h and a. In this problem, you do not need to use a for anything. (Though I would expect everyone to be able to find an equation relating h, r and a.)
5. The region R is bounded by the graph of y = x2 and the line y = 4.
a. Find the area of R.
b. Find the volume of the solid generated by revolving R about the x-axis.
c. There exists a number k, k > 4, such that when R is revolved about the line y = k, the resulting solid has the same solid as the volume in part b. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k.
AP Calculus Areas and Volumes - 7
1. Find the volume of a solid if the base is a circular disk of radius r and cross sections perpendicular to a diameter are squares.
2. Find the volume of a solid if the base is the parabolic region given by [pic] and cross sections perpendicular to the y-axis are equilateral triangles. (Note: You should be able to figure out the area of an equilateral triangle of side s but I will save you the time: it is [pic].)
3. The ellipse [pic] forms the base of a solid. For this solid, cross-sections perpendicular to the x-axis are isosceles triangles with bases in the x-y plane and heights equal to k times their base. Find the value of k so that the volume of the solid is 30.
4. Find the volume of a solid if the base is bounded by the graphs of y = 8x – x2 and [pic]and cross sections perpendicular to the x-axis are semicircles with their diameters in the base of the solid.
5. A solid has a circular base of radius 1, centered at the origin. Cross sections perpendicular to the x-axis are rectangles with the height at any value of x given by h = 1 – x2. Find the volume of this solid.
AP Calculus Review: Area and Volume
1. Let f be a continuous function that is always positive and decreasing on the interval [a, b] (where 0 < a < b), let b be a root of f and let f -1 be the inverse of f .
a. Write an expression for the area A bounded by the graph of f, the x-axis and the lines x = a and x = b.
b. Write an expression for the average value of f on the interval [a, b].
c. Write an expression for the volume that results when the area A is rotated about the x-axis.
d. Write an expression in one variable for the volume that results when the area A is rotated about the
y-axis.
e. Let the region R be the base of a solid having cross sections perpendicular to the y-axis in the shape of squares. Write an expression for the volume of this solid.
2. Let the region R be enclosed by the graphs of [pic], the coordinate axes and the line x = 8.
a. Find the area of region R.
b. The line x = k divides the region into two regions of equal area. Find the value of k.
c. Find the volume that results when R is rotated about the line y = 2.
d. Suppose the region R is rotated about the x-axis. A plane perpendicular to the x-axis at x = b divides the resulting volume into two equal parts. Find the value of b.
3. The interior of a decorative vase is 1.9 feet tall and has a radius that varies with its height according to the function r = h3 – 3h2 + 2h + 1.
a. Find the volume of the vase.
b. Set up an integral equation that could be used to find the depth D of water in the vase when it is half full. It is not necessary to solve for D.
(This assignment continued on the next page.)
4. The region R is bounded by the graph of [pic] and the line y = 2. Set up appropriate integrals to find the following (it is not necessary to evaluate all the integrals).
a. The area of R.
b. The volume that results if R is rotated about the x-axis.
c. The volume that results if the portion of R in the first quadrant is rotated about the y-axis.
d. The volume that results when R is rotated about the lien y = 20.
e. The volume of the solid having R as its base and with cross-sections perpendicular to the x-axis being rectangles with height x + 3.
5. The base of a solid is bounded in the x-y plane by the graphs of [pic]and [pic]and the lines x = 0 and x = 4. Cross sections taken perpendicular to the x-axis are semi-circles with a diameter in the x-y plane. Find the volume of the solid.
6. Set up appropriate integrals to find the volume obtained by rotating the region bounded by the graphs of [pic], y = 2e-x/2, and the y-axis about
a. the x-axis and b. the y-axis.
7. You need to find the area enclosed by the graph of [pic], the tangent line to f at x = 5, and the coordinate axes. There are (at least) two reasonable ways to do this using two integrals. How many ways can you find to do it with just one integral?
-----------------------
x
y
[pic]
[pic]
y =x + 5
y = -1
y =2
x
y
[pic]
[pic]
[pic]
y = sec2x
y = 2
r
h
r
h
r
h
a
r
h
................
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