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Calculus - Applications of the Definite Integral

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    Study Guide Calculus – Applications of the Definite Integral 1. Volumes of Solids of Revolution One powerful use of definite integrals is finding the volume of a solid of revolution . These solids are formed when a flat (plane) region is rotated 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 three methods: • Disk method • Washer method • Cylindrical shell method In this chapter, we focus on the disk method and the washer method . Each method uses a different type of cross - section, so the setup of the integral will change slightly. The Disk Method When do we use the disk method? Use the disk method when: • The axis of rotation is a boundary of the region, and • The cross sections are taken perpendicular to the axis of rotation. In this situation, each cross section looks like a solid 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 all the slices using an integral.

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    Study Guide Disk Method Formulas • If the disk is perpendicular to the x - axis , the radius must be written as a function of x . • If the disk is perpendicular to the y - axis , the radius must be written as a function of y . Rotating around the x - axis If 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 - axis If 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 the radius of the disk , which is the distance from the curve to the axis of rotation. Example 1 : Using the Disk Method Problem Find the volume of the solid formed by revolving the region bounded by (y = x 2 ) and the x - axis on the interval ([ - 2, 3]) about the x - axis. Step 1: Choose the method Because the x - axis is a boundary of the region , the disk method applies.

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

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    Study Guide Final Answer: The Washer Method When do we use the washer method? Use the washer method when: • The axis of rotation is not a boundary of the region, and • The cross sections are taken perpendicular to the axis of rotation. In this case, each cross section looks like a washer (a disk with a hole in the center). Understanding washers A washer has: • An outer radius (R) • An inner radius (r) The area of a washer is: To find the volume, multiply this area by the thickness and integrate. Washer Method Formulas Rotating around the x - axis

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    Study Guide If 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 - axis If 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 the outer radius • (g) represents the inner radius • Both radii are measured as distances from the axis of rotation Example 2 : Using the Washer Method Problem Find the volume of the solid formed by revolving the region bounded by about the x - axis .

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

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    Study Guide • The upper curve is (y = x + 4) • The lower curve is (y = x 2 + 2) Because the x - axis is not a boundary of the region, we must use the washer method . Step 3: Set up the integral For washers: • Outer radius: (R = x + 4) • Inner radius: (r = x 2 + 2) The washer formula is: Substitute the values: Step 4: Simplify and evaluate Expand each expression: Subtract: Integrate:

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    Study Guide Evaluate: Final Answer The Cylindrical Shell Method When do we use the shell method? Use the cylindrical shell method when: • The cross sections are taken parallel to the axis of rotation. Instead of disks or washers, we imagine peeling the solid into thin cylindrical shells . Understanding a cylindrical shell Each 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 Formulas Rotating around the y - axis

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    Study Guide If 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 - axis If 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 • The radius ((x) or (y)) represents the distance from the shell to the axis of rotation. • The function (f(x)) or (f(y)) represents the height of the shell. • Shell method is often helpful when washer or disk methods become complicated. Example 3: Using the Cylindrical Shell Method Problem Find the volume of the solid formed by revolving the region bounded by and the x - axis on the interval ([1, 3]) about the y - axis .

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    Study Guide Figure 3 Diagram for Example 3. Step 1: Choose the correct method Because the solid is being rotated about the y - axis (a vertical axis), and the slices will be taken parallel to that axis, the cylindrical shell method is the best choice. Step 2: Understand the shells Using the shell method, we imagine the region made up of many thin vertical strips. When each strip is revolved around the y - axis, it forms a cylindrical shell . For each shell: • Radius = distance from the shell to the y - axis = (x) • Height = value of the function = (f(x) = x 2 ) • Thickness = (dx)

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