Presentation on theme: "Applications of Integration"— Presentation transcript:
1Applications of Integration In this chapter we explore some of the applications of thedefinite integral by using it forComputing the area between curvesComputing the volumes of solidsComputing the work done by a varying forceComputing average value of a functionThe common theme is the following general method, which is similar to the one we used to find areas under curves:We break up a Q quantity into a large number of small parts. We next approximate each small part by a quantity of the form and thus approximate Q by a Riemann sum. Then we take the limit and express Q as an integral Finally we evaluate the integral using the Fundamental Theorem of Calculus or the Midpoint Rule.
13Some regions are best treated by regarding x as a function of y
14If we try vertical strips, we have to integrate in two parts: We can find the same area using a horizontal strip.Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.
15We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.length of stripwidth of strip
16p General Strategy for Area Between Curves: 1 Sketch the curves. Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.)23Write an expression for the area of the strip.(If the width is dx, the length must be in terms of x.If the width is dy, the length must be in terms of y.4Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.)5Integrate to find area.p
183Find the volume of the pyramid:Consider a horizontal slice through the pyramid.The volume of the slice is s2dh.If we put zero at the top of the pyramid and make down the positive direction, then s=h.hsThis correlates with the formula:dh3
19Method of Slicing:1Sketch the solid and a typical cross section.Find a formula for V(x).(Note that I used V(x) instead of A(x).)23Find the limits of integration.4Integrate V(x) to find volume.
20h 45o x A 45o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge.You could slice this wedge shape several ways, but the simplest cross section is a rectangle.xyIf we let h equal the height of the slice then the volume of the slice is:xh45oSince the wedge is cut at a 45o angle:Since
21Even though we started with a cylinder, p does not enter the calculation! xy
22p Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume.Identical Cross Sectionsp
27Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape.So I put a piece of wood in a lathe and turn it to a shape to match the curve.
28r= the y value of the function How could we find the volume of the cone?One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes.The volume of each flat cylinder (disk) is:In this case:r= the y value of the functionthickness = a small change in x = dx
29The volume of each flat cylinder (disk) is: If we add the volumes, we get:
30This application of the method of slicing is called the disk method This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk.If the shape is rotated about the x-axis, then the formula is:Since we will be using the disk method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes.A shape rotated about the y-axis would be:
31Revolving a Function Consider a function f(x) on the interval [a, b] Now consider revolving that segment of curve about the x axisWhat kind of functions generated these solids of revolution?f(x)ab
32Disksf(x)We seek ways of using integrals to determine the volume of these solidsConsider a disk which is a slice of the solidWhat is the radiusWhat is the thicknessWhat then, is its volume?dx
33DisksTo find the volume of the whole solid we sum the volumes of the disksShown as a definite integralf(x)ab
34y-axis is revolved about the y-axis. Find the volume. The region between the curve , and they-axis is revolved about the y-axis. Find the volume.We use a horizontal disk.yxThe thickness is dy.The radius is the x value of the functionvolume of disk
36The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis:The volume can be calculated using the disk method with a horizontal disk.
37The region bounded byand is revolved about the y-axis.Find the volume.If we use a horizontal slice:The “disk” now has a hole in it, making it a “washer”.The volume of the washer is:outerradiusinnerradius
38This application of the method of slicing is called the washer method This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.The washer method formula is:Like the disk method, this formula will not be on the formula quizzes. I want you to understand the formula.
39p If the same region is rotated about the line x=2: The outer radius is:The inner radius is:rRp
40Georgia Aquarium, Atlanta Find the volume of the region bounded by , , and revolved about the y-axis.We can use the washer method if we split it into two parts:innerradiuscylinderouterradiusthicknessof sliceJapanese Spider CrabGeorgia Aquarium, Atlanta
47The solids in Examples 1–5 are all called solids of revolution because they are obtained by revolving a region about a line. In general, we calculate the volume of a solid of revolution by using the basic defining formulaWe find the cross-sectional area in one of the following ways:
49cross section Here is another way we could approach this problem: If we take a vertical sliceand revolve it about the y-axiswe get a cylinder.If we add all of the cylinders together, we can reconstruct the original object.
50r is the x value of the function. h is the y value of the function. cross sectionThe volume of a thin, hollow cylinder is given by:r is the x value of the function.h is the y value of the function.thickness is dx.
51This is called the shell method because we use cylindrical shells. cross sectionIf we add all the cylinders from the smallest to the largest:
52Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.
53If we take a vertical slice and revolve it about the y-axiswe get a cylinder.Shell method:This model of the shell method and other calculus models are available from: Foster Manufacturing Company, 1504 Armstrong Drive, Plano, Texas Phone/FAX: (972)
54Note:When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.
55Shell Method Based on finding volume of cylindrical shells Add these volumes to get the total volumeDimensions of the shellRadius of the shellThickness of the shellHeight
56The Shell Consider the shell as one of many of a solid of revolution The volume of the solid made of the sum of the shellsdxf(x)f(x) – g(x)xg(x)
57Try It Out!Consider the region bounded by x = 0, y = 0, and
58Hints for Shell Method Sketch the graph over the limits of integration Draw a typical shell parallel to the axis of revolutionDetermine radius, height, thickness of shellVolume of typical shellUse integration formula
59Rotation About x-AxisRotate the region bounded by y = 4x and y = x2 about the x-axisWhat are the dimensions needed?radiusheightthicknessthickness = dyradius = y
60Rotation About Noncoordinate Axis Possible to rotate a region around any lineRely on the basic concept behind the shell methodf(x)g(x)x = a
61Rotation About Noncoordinate Axis What is the radius?What is the height?What are the limits?The integral:rf(x)g(x)a – xx = cx = af(x) – g(x)c < x < a
62Try It OutRotate the region bounded by 4 – x2 , x = 0 and, y = 0 about the line x = 2Determine radius, height, limitsr = 2 - x4 – x2
74Average value of a function The average value of function f on the interval [a, b] is defined asNote: For a positive function, we can think of this definition as saying area/width = average heightExample: Find the average value of f(x)=x3 on [0,2].
77The Mean Value Theorem for Integrals Theorem: If f is continuous on [a, b], then there exists a number c in [a, b] such thatExample: Find c such that fave=f(c) for f(x)=x3 on [0,2].From previous slide, f(c)=fave=2.Thus, c3=2, so