Integration to Compute Volumes, Work. Goals: Volumes by Slicing Volumes by Cylindrical Shells Work

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Lambert Byrd
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1 Week #8: Integration to Compute Volumes, Work Goals: Volumes by Slicing Volumes by Cylindrical Shells Work 1

2 Volumes by Slicing - 1 Volumes by Slicing In the integration problems considered in this section the accumulated total is a volume, rather than an area. In each of the examples, this total is obtained by regarding the solid figure as a stack of infinitely thin slices. We will construct integrals for these volumes using the method of slices. If the slices have a simple shape (say if they are circular so that they can be thought of as infinitely thin cylinders) so that we can obtain a simple formula for the volume of a typical slice, then we can add (that is, integrate) the volumes of these slices to get a total volume for the figure.

3 Volumes by Slicing - 2 Problem. Find the volume of the solid obtained by rotating the triangle bounded by x = 0, y = 0 and x + y = 1 about the x-axis.

4 Volumes by Slicing - 3

5 Volumes by Slicing - 4

6 Volumes by slicing Choose a direction to slice Volumes by Slicing - Examples - 1 Find a formula for the volume of one slice Create sum/integral to compute total volume

7 Volumes by Slicing - Examples - 2 Problem. Find the volume of the solid obtained by rotating the region bounded by x = 0 and x = y y 2 about the y-axis.

8 Volumes by Slicing - Examples - 3 Problem continued.

9 Volumes by Slicing - Examples - 4 Problem. Suppose a vase is such that when you fill it with water up to depth h (measured in cm) the surface area of the water in the vase is A(h) square centimeters. Then if you fill the vase up to 30 cm, the integral that is equal to the volume of water in the vase is A h A(h) dh B. C π A(h) dh A (h) dh D A(h) dh

10 Volumes by Slicing - Examples 2-1 Problem. Find the volume of the solid obtained by rotating the region bounded by y = x 3 and x = y 2 about the x-axis.

11 Volumes by Slicing - Examples 2-2 Problem continued.

12 Problem. Find the volume of a ball of radius r. Volumes by Slicing - Examples 2-3

13 Volumes by Slicing - Examples 2-4 Problem continued.

14 Volumes by Cylindrical Shells - 1 Volumes by Cylindrical Shells In this kind of application of integration, the accumulated total is a volume consisting of infinitely many infinitely thin concentric cylindrical shells. The goal is to find an expression for the volume of a typical shell, and then to add (that is, integrate) these volumes to get the total. (Note that Example done using washers can also be done using cylindrical shells.)

15 Volumes by Cylindrical Shells - 2 Problem. Find the volume of the solid produced by rotating the region bounded by y = x and y = 4x(1 x) about the y-axis.

16 Volumes by Cylindrical Shells - 3 Problem continued.

17 Volumes by Cylindrical Shells - 4 Problem continued.

18 Volumes by Cylindrical Shells - Examples - 1 Problem. Suppose we have a function f(x) that is positive on the interval [1, 2] and suppose that D is the region between the graph of f, the lines x = 1, x = 2, and the x-axis. D 1 2 If we rotate the region D about the y-axis to form a solid S, which of the following integrals represents the volume of S? 2 2 A. 1 2πf(x) dx B. 1 2π x f(x) dx 2 1 C. 1 π x 2 f(x) dx D. 0 π x 2 f(x) dx

19 Volumes by Cylindrical Shells - Examples - 2 Problem. Find the volume generated by rotating about the line x = 1 the region that lies in the first quadrant and is bounded by y = x 2, y = 4 and x = 0.

20 Volumes by Cylindrical Shells - Examples - 3 Problem continued.

21 Work - 1 Work The basic formula for work is the product of force times distance. If force is measured in Newtons and distance in meters, the answer is in Joules.

22 Work - 2 Problem. If you pull an object, using a force of 5 N, and you move it from x = 0 to x = 15 m, how much work have you done? Give units in your answer.

23 Work - 3 Problem. If the force you applied was changing as you pulled, with F = (5 0.1x) N, how would this affect how you can calculate the total work?

24 Work - 4 Problem continued.

25 Work - Examples - 1 Problem. ( When a particle is x meters from the origin, a force measuring cos N acts on it. How much work is done by moving πx ) 3 the particle from x = 1 to x = 2?

26 Work - Aquarium Problem - 1 Problem. An aquarium 2 m long, 1 m wide and 1 m deep is full of water. Find the minimum amount of work needed to pump half of the water out of the aquarium.

27 Work - Aquarium Problem - 2 Problem continued.

28 Work - Aquarium Problem - 3 Comments on the aquarium problem (1) Strictly speaking, the assumptions behind the problem are highly idealized. In any real situation the water would come out of the hose with some amount of kinetic energy, and this extra energy adds to the work done. To calculate the minimum amount of work required is to ignore these effects. Even if in real life the amount of work required is always somewhat more than what this calculation tells, it is nevertheless helpful to know that it gives the absolute minimum that could be reached. (2) The method used really hinges on the conservation of energy: energy gained = work done. We calculated this work by calculating the increase in (potential) energy in the horizontal slabs of water. (3) To minimize the work needed, we imagine the pumping done slowly so no kinetic energy is created.

29 Work - Aquarium Problem - 4 (4) In principle, if we knew what happened to each particle of water, we could do a more detailed and realistic analysis. It would require knowing where the hose is placed (on the bottom of the aquarium or higher) and a calculation of the work done on or by each individual water particle as it is pushed down the tube and then up again, or (in other cases) as it sinks closer to the bottom of the aquarium. In practice this picture becomes far too complicated to use. The power of the principle of energy conservation is in its ability to simplify the problem.

30 Work - Lake Problem - 1 Problem. A large cylindrical tank is filled with water. There is a drain in the center of the bottom of the tank, two meters above the surface of a lake. A hose is attached to the drain, and the tank is allowed to empty through the hose onto the surface of the lake. We want to calculate the loss of potential energy of the water as it runs from the tank to the surface of the lake.

31 Work - Lake Problem - 2 Problem. How should we choose the slices of water for our integral, and why? A. Horizontal slabs because it worked last time. B. Horizontal slabs because all the points in a horizontal slab are the same distance above the surface of the lake. C. Horizontal slabs because when it is at rest, water surface is always horizontal. D. Cylindrical shells because the tank is cylindrical. E. Cylindrical shells because each such shell is at a constant radius from the center, where the drain is located.

32 Work - Parabolic Tank Problem - 1 Problem. The parabola y = x 2 is rotated about the y-axis, and filled with water to the level y = 3. How much work is required to pump the water out through a hole located at y = 4? (Assume all scales are in meters.)

33 Work - Parabolic Tank Problem - 2 Problem continued.

34 Angular Momentum - 1 Problem. The angular momentum of a point mass rotating about an axis at ω radians per second is defined as the product of the mass, the square of the distance from the centre, and the angular velocity ω (that is, mr 2 ω). Suppose we have a large metal cylinder of uniform density rotating about its axis. If we want to use an integral to calculate the total angular momentum of the cylinder we have to think of the cylinder in terms of a parametric family of pieces chosen in such a way that the angular momentum of each piece is easily calculated. How should we do that, and why?

35 Angular Momentum - 2 A. Horizontal slices of vertical thickness dy, because the volumes of these slices are given by π r 2 dy B. Horizontal slices of vertical thickness dy, because the cylinder is rotationally symmetric C. Cylindrical shells because the object is a cylinder D. Cylindrical shells because on each such shell the radius to the axis is the same at every point, and thus the angular momentum of such a shell is easily calculated.

36 Discussion Angular Momentum - 3

For the intersections: cos x = 0 or sin x = 1 2

For the intersections: cos x = 0 or sin x = 1 2 Chapter 6 Set-up examples The purpose of this document is to demonstrate the work that will be required if you are asked to set-up integrals on an exam and/or quiz.. Areas () Set up, do not evaluate, any