Volumes of Solids of Revolution

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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.**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.**The four examples in the Curriculum Module use the disk**method or the washer method.**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**Example 1**• Think about each of the lines spinning and creating the solid. • Draw one representative disk. • Draw in the radius.**Example 1**• Find the radius of the larger circle, its area and the volume of the disk.**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.**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.**Example 1**• Find the radius of the smaller circle, its area and the volume of the disk.**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.**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)**Example 1**• The final volume will be the difference between the two volumes.**Example 2 Line of Rotation Above the Region to be Rotated**• Rotate the same region about y = 2 • Notice that**Example 2 Line of Rotation Above the Region to be Rotated**• The area of the larger circle is**Example 2 Line of Rotation Above the Region to be Rotated**• The sum of the volumes is**Example 2 Line of Rotation Above the Region to be Rotated**• The area of the smaller circle is**Example 2 Line of Rotation Above the Region to be Rotated**• The sum of the volumes is**Example 2 Line of Rotation Above the Region to be Rotated**• The volume of the solid is the difference between the two volumes**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**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.**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 + (lny) Area of the larger circle is**Example 3 Line of Rotation to the Left of the Region to be**Rotated • Volume of each disk:**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 + (y2 – 2) • Area of the larger circle is**Example 3 Line of Rotation to the Left of the Region to be**Rotated • Volume of each disk:**Example 3 Line of Rotation to the Left of the Region to be**Rotated • Difference in the volume is**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**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**Example 4 Line of Rotation to the Right of the Region to Be**Rotated • Area of Larger disk: • The volume of the disk is**Example 4 Line of Rotation to the Right of the Region to Be**Rotated • Volume of all the disks are**Example 4 Line of Rotation to the Right of the Region to Be**Rotated • Area of smaller disk: • The volume of the disk is**Example 4 Line of Rotation to the Right of the Region to Be**Rotated • Volume of all the disks are**Example 4 Line of Rotation to the Right of the Region to Be**Rotated • Find the difference in the volumes