volumes by slicing volumes by cylindrical shells workapsc171/notes/notes08.pdf · 2019. 11. 14. ·...

Block #8: Integration to Compute Volumes, Work

Goals:

•Volumes by Slicing•Volumes by Cylindrical Shells•Work

1

Volumes by Slicing - 1

Volumes by Slicing

In the integration problems considered in this section the accumulatedtotal is a volume, rather than an area. In each of the examples, thistotal is obtained by regarding the solid figure as a stack of infinitelythin 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 theycan be thought of as infinitely thin cylinders) so that we can obtaina 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 volumefor the figure.


Problem. Find the volume of the solid obtained by rotating thetriangle bounded by x = 0, y = 0 and x + y = 1 about the x-axis.

Volumes by Slicing - Examples - 1

Volumes by slicing

• Choose a direction to slice

• Find a formula for the volume of one slice

• Create sum/integral to compute total volume


Problem. Find the volume of the solid obtained by rotating theregion bounded by x = 0 and x = y − y2 about the y-axis.


Problem continued.


Problem. Suppose a vase is such that when you fill it with waterup to depth h (measured in cm) the surface area of the water in thevase is A(h) square centimeters. Then if you fill the vase up to 30cm, the integral that is equal to the volume of water in the vase is

A.

∫ 30

0h A(h) dh

B.

∫ 30

02π A(h) dh

C.

∫ 30

0A′(h) dh

D.

∫ 30

0A(h) dh

Volumes by Slicing - Examples 2 - 1

Problem. Find the volume of the solid obtained by rotating theregion bounded by y = x3 and x = y2 about the x-axis.


Problem continued.


Problem. Find the volume of a ball of radius r.


Problem continued.

Volumes by Cylindrical Shells - 1

Volumes by Cylindrical Shells

In this kind of application of integration, the accumulated total is avolume consisting of infinitely many infinitely thin concentric cylin-drical shells. The goal is to find an expression for the volume of atypical shell, and then to add (that is, integrate) these volumes toget the total. (Note that Example done using washers can also bedone using cylindrical shells.)


Problem. Find the volume of the solid produced by rotating theregion bounded by y = x and y = 4x(1− x) about the y-axis.


Problem continued.

Volumes by Cylindrical Shells - Examples - 1

Problem. Suppose we have a function f (x) that is positive on theinterval [1, 2] and suppose that D is the region between the graph off , 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, whichof the following integrals represents the volume of S?

A.

∫ 2

12πf (x) dx B.

∫ 2

12π x f (x) dx

C.

∫ 2

1π x2 f (x) dx D.

∫ 1

0π x2 f (x) dx


Problem. Find the volume generated by rotating about the linex = −1 the region that lies in the first quadrant and is bounded byy = x2 , y = 4 and x = 0 .


Problem continued.

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 answeris in Joules.

Work - 2

Problem. If you pull an object, using a force of 5 N, and you moveit from x = 0 to x = 15 m, how much work have you done? Giveunits in your answer.

Work - 3

Problem. If the force you applied was changing as you pulled, withF = (5 − 0.1x) N, how would this affect how you can calculate thetotal work?

Work - 4

Problem continued.

Work - Examples - 1

Problem. When a particle is x meters from the origin, a force mea-

suring cos(πx

3

)N acts on it. How much work is done by moving

the particle from x = 1 to x = 2 ?

Work - Aquarium Problem - 1

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


Problem continued.


Comments on the aquarium problem

(1) Strictly speaking, the assumptions behind the problem are highlyidealized. In any real situation the water would come out of thehose with some amount of kinetic energy, and this extra energyadds to the work done. To calculate the minimum amount ofwork required is to ignore these effects. Even if in real life theamount of work required is always somewhat more than what thiscalculation tells, it is nevertheless helpful to know that it givesthe absolute minimum that could be reached.

(2) The method used really hinges on the conservation of energy: en-ergy gained = work done. We calculated this work by calculatingthe 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.


(4) In principle, if we knew what happened toeach particle of water, we could do a moredetailed and “realistic” analysis. It would re-quire knowing where the hose is placed (onthe bottom of the aquarium or higher) anda calculation of the work done on or by eachindividual water particle as it is pushed downthe tube and then up again, or (in other cases)as it sinks closer to the bottom of the aquar-ium.In practice this picture becomes far too com-plicated to use. The power of the principle ofenergy conservation is in its ability to simplifythe problem.

Work - Lake Problem - 1

Problem. A large cylindrical tank is filled with water. There is adrain in the center of the bottom of the tank, two meters above thesurface of a lake. A hose is attached to the drain, and the tank isallowed to empty through the hose onto the surface of the lake. Wewant to calculate the loss of potential energy of the water as it runsfrom the tank to the surface of the lake.

Work - Lake Problem - 2

Problem. How should we choose the “slices” of water for our inte-gral, and why?

A. Horizontal slabs because it worked last time.

B. Horizontal slabs because all the points in a horizontal slab are thesame distance above the surface of the lake.

C. Horizontal slabs because when it is at rest, water surface is alwayshorizontal.

D. Cylindrical shells because the tank is cylindrical.

E. Cylindrical shells because each such shell is at a constant radiusfrom the center, where the drain is located.

Work - Parabolic Tank Problem - 1

Problem. The parabola y = x2 is rotated about the y-axis, andfilled with water to the level y = 3. How much work is required topump the water out through a hole located at y = 4? (Assume allscales are in meters.)

Work - Parabolic Tank Problem - 2

Problem continued.

Angular Momentum - 1

Problem. The angular momentum of a point mass rotating aboutan 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, mr2ω). Suppose we have a large metal cylinder of uniformdensity rotating about its axis. If we want to use an integral tocalculate the total angular momentum of the cylinder we have tothink of the cylinder in terms of a parametric family of “pieces”chosen in such a way that the angular momentum of each piece iseasily calculated. How should we do that, and why?


A. Horizontal slices of vertical thickness dy, because the volumes ofthese slices are given by π r2 dy

B. Horizontal slices of vertical thickness dy, because the cylinder isrotationally symmetric

C. Cylindrical shells because the object is a cylinder

D. Cylindrical shells because on each such shell the radius to the axisis the same at every point, and thus the angular momentum ofsuch a shell is easily calculated.


Discussion

volumes by slicing volumes by cylindrical shells workapsc171/notes/notes08.pdf · 2019. 11. 14. ·...

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