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VOLUMES OF SOLIDS OF REVOLUTION

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Questions involving the area of a region between curves, and the volume of a solid formed when this region is rotated about a horizontal or vertical line, appear regularly on both the AP Calculus AB and BC exams. Students have difficulty when the solid is formed by use a line of rotation other than the x- or y-axis.

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These types of volume are part of the type of volume problems students must solve on the AP test. Students should find the volume of a solid with a known cross section. The Shell method is not part of the AB or the BC course of study anymore.

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The four examples in the Curriculum Module use the disk method or the washer method.

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Example 1 Line of Rotation Below the Region to be Rotated Picture the solid (with a hole) generated when the region bounded by and are revolved about the line y = -2. First find the described region Then create the reflection over the line y=-2

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Example 1 Think about each of the lines spinning and creating the solid. Draw one representative disk. Draw in the radius.

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Example 1 Find the radius of the larger circle, its area and the volume of the disk.

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Sum up these cylinders to find the total volume The larger the number of disks and the thinner each disk, the smoother the stack of disks will be. To obtain a perfectly smooth solid, we let n approach infinity and Δx approach 0.

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The points of intersection can be found using the calculator. Store these in the graphing calculator (A=-1.980974,B=0.13793483) (C=0.44754216,D=1.5644623) Write an integral to find the volume of the solid.

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Example 1 Find the radius of the smaller circle, its area and the volume of the disk.

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Sum up these cylinders to find the total volume The larger the number of disks and the thinner each disk, the smoother the stack of disks will be. To obtain a perfectly smooth solid, we let n approach infinity and Δx approach 0.

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Using the points of intersection write a second integral for the inside volume. (A=-1.980974,B=0.13793483) (C=0.44754216,D=1.5644623)

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Example 1 The final volume will be the difference between the two volumes.

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Example 2 Line of Rotation Above the Region to be Rotated Rotate the same region about y = 2 Notice that

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Example 2 Line of Rotation Above the Region to be Rotated The area of the larger circle is

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Example 2 Line of Rotation Above the Region to be Rotated The sum of the volumes is

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Example 2 Line of Rotation Above the Region to be Rotated The area of the smaller circle is

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Example 2 Line of Rotation Above the Region to be Rotated The sum of the volumes is

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Example 2 Line of Rotation Above the Region to be Rotated The volume of the solid is the difference between the two volumes

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Example 3 Line of Rotation to the Left of the Region to be Rotated Line of Rotation: x = -3 Use the same two functions Create the reflection Draw the two disks and mark the radius

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Example 3 Line of Rotation to the Left of the Region to be Rotated The radius will be an x- distance so we will have to write the radius as a function of y.

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Example 3 Line of Rotation to the Left of the Region to be Rotated The radius of the larger disk is 3 + the distance from the y-axis or 3 + (ln y) Area of the larger circle is

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Example 3 Line of Rotation to the Left of the Region to be Rotated Volume of each disk:

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Example 3 Line of Rotation to the Left of the Region to be Rotated The radius of the smaller disk is 3+ the distance from the y-axis or 3 + (y 2 – 2) Area of the larger circle is

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Example 3 Line of Rotation to the Left of the Region to be Rotated Volume of each disk:

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Example 3 Line of Rotation to the Left of the Region to be Rotated Difference in the volume is

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Example 4 Line of Rotation to the Right of the Region to Be Rotated Line of Rotation: x = 1 Create the region, reflect the region and draw the disks and the radius

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Example 4 Line of Rotation to the Right of the Region to Be Rotated Notice the larger radius is 1 + the distance from the y-axis to the outside curve. The distance is from the y-axis is negative so the radius is

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Example 4 Line of Rotation to the Right of the Region to Be Rotated Area of Larger disk: The volume of the disk is

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Example 4 Line of Rotation to the Right of the Region to Be Rotated Volume of all the disks are

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Example 4 Line of Rotation to the Right of the Region to Be Rotated Area of smaller disk: The volume of the disk is

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Example 4 Line of Rotation to the Right of the Region to Be Rotated Volume of all the disks are

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Example 4 Line of Rotation to the Right of the Region to Be Rotated Find the difference in the volumes

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