Calculus - Applications of the Definite Integral - Quick Guides

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Study GuideCalculus–Applications of the Definite Integral1.Volumes of Solids of RevolutionOne powerful use of definite integrals is finding thevolume of a solid of revolution.These solids are formed when a flat (plane) region isrotated around a horizontal or vertical line.Depending on how the region is rotated and where the axis of rotation is located, we use one of threemethods:•Disk method•Washer method•Cylindrical shell methodIn this chapter, we focus on thedisk methodand thewasher method. Each method uses a differenttype of cross-section, so the setup of the integral will change slightly.The Disk MethodWhen do we use the disk method?Use thedisk methodwhen:•Theaxis of rotation is a boundaryof the region, and•The cross sections are takenperpendicularto the axis of rotation.In this situation, each cross section looks like asolid disk(a filled-in circle).Why is it called the disk method?Each cross section is a circle.The area of a circle is:To find volume, we multiply this area by a very small thickness (either (dx) or (dy)) and then add up allthe slices using an integral.

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Study GuideDisk Method Formulas•If the disk isperpendicular to the x-axis, the radius must be written as a function ofx.•If the disk isperpendicular to the y-axis, the radius must be written as a function ofy.Rotating around the x-axisIf the region bounded by (y = f(x)) and the x-axis on the interval ([a, b]) is revolved about the x-axis,the volume is:Rotating around the y-axisIf the region bounded by (x = f(y)) and the y-axis on the interval ([a, b]) is revolved about the y-axis,the volume is:In both cases, the function represents theradius of the disk, which is the distance from the curve tothe axis of rotation.Example1:Using the Disk MethodProblemFind the volume of the solid formed by revolving the region bounded by(y = x2) and the x-axis on the interval ([-2, 3]) about the x-axis.Step 1: Choose the methodBecause thex-axis is a boundary of the region, thedisk methodapplies.

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Study GuideFigure 1 Diagram for Example 1.Step 2: Set up the integralThe radius of each disk is:Using the disk formula:Simplify:Step 3: Evaluate the integral

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Study GuideFinal Answer:The Washer MethodWhen do we use the washer method?Use thewasher methodwhen:•Theaxis of rotation is not a boundaryof the region, and•The cross sections are takenperpendicularto the axis of rotation.In this case, each cross section looks like awasher(a disk with a hole in the center).Understanding washersA washer has:•Anouter radius(R)•Aninner radius(r)The area of a washer is:To find the volume, multiply this area by the thickness and integrate.Washer Method FormulasRotating around the x-axis

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Study GuideIf the region bounded by (y = f(x)) and (y = g(x)) on ([a, b]), where (f(x)≥g(x)), is revolved about the x-axis:Rotating around the y-axisIf the region bounded by (x = f(y)) and (x = g(y)) on ([a, b]), where (f(y)≥g(y)), is revolved about the y-axis:Here:•(f) represents theouter radius•(g) represents theinner radius•Both radii are measured as distances from the axis of rotationExample 2:Using the Washer MethodProblemFind the volume of the solid formed by revolving the region bounded byabout thex-axis.

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Study GuideFigure 2 Diagram for Example 2.Step 1: Find the points of intersectionTo determine the limits of integration, set the two equations equal:Rewriting:Factor:So, the curves intersect at:Step 2: Identify the outer and inner curvesOn the interval ([-1, 2]):

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Study Guide•Theupper curveis (y = x + 4)•Thelower curveis (y = x2+ 2)Because the x-axis isnot a boundaryof the region, we must use thewasher method.Step 3: Set up the integralFor washers:•Outer radius: (R = x + 4)•Inner radius: (r = x2+ 2)The washer formula is:Substitute the values:Step 4: Simplify and evaluateExpand each expression:Subtract:Integrate:

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Study GuideEvaluate:Final AnswerThe Cylindrical Shell MethodWhen do we use the shell method?Use thecylindrical shell methodwhen:•The cross sections are takenparallelto the axis of rotation.Instead of disks or washers, we imagine peeling the solid intothin cylindrical shells.Understanding a cylindrical shellEach shell has:•Radius(r): distance from the shell to the axis of rotation•Height(h): length of the shell•Thickness: a very small amount ((dx) or (dy))The volume of one shell is:Shell Method FormulasRotating around the y-axis

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Study GuideIf the region bounded by (y = f(x)) and the x-axis on ([a, b]), where (f(x)≥0), is revolved about the y-axis:Rotating around the x-axisIf the region bounded by (x = f(y)) and the y-axis on ([a, b]), where (f(y)≥0), is revolved about the x-axis:Important Notes•Theradius((x) or (y)) represents the distance from the shell to the axis of rotation.•The function (f(x)) or (f(y)) represents theheightof the shell.•Shell method is often helpful when washer or disk methods become complicated.Example 3:Using the Cylindrical Shell MethodProblemFind the volume of the solid formed by revolving the region bounded byand the x-axis on the interval ([1, 3]) about they-axis.

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Study GuideFigure 3 Diagram for Example 3.Step 1: Choose the correct methodBecause the solid is being rotated about they-axis(a vertical axis), and the slices will be takenparallelto that axis, thecylindrical shell methodis the best choice.Step 2: Understand the shellsUsing the shell method, we imagine the region made up of many thin vertical strips. When each stripis revolved around the y-axis, it forms acylindrical shell.For each shell:•Radius= distance from the shell to the y-axis = (x)•Height= value of the function = (f(x) = x2)•Thickness= (dx)

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