Unit #13 : Integration to Find Areas and Volumes, Volumes of Revolution
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1 Unit #13 : Integration to Find Areas and Volumes, Volumes of Revolution Goals: Beabletoapplyaslicingapproachtoconstructintegralsforareasandvolumes. Be able to visualize surfaces generated by rotating functions around different axes. Reading: Sections 8.1, 8.2
2 Integrals as Areas - Review - 1 Integrals as Areas - Review Reading: Section 8.1 Example: Write the integral that represents the area underneath the graph y = x 2 between x = 0 and x = 2. Evaluate the integral to find the area.
3 Integrals as Areas - Review - 2 Illustrate how this shape can be constructed by accumulating small rectangles of varying heights.
4 Integrals as Areas - Review - 3 Now show how the exact same area can be constructed by using horizontal rectangles. Write the analogous intervals, widths, etc. on the diagram.
5 Integrals as Areas - Review - 4 Write out first a sum, then a new integral, that would represent the exact area in the sketch. Evaluate the integral.
6 Integrals as Sums of Slices - 1 Integrals as Sums of Slices Most people visualize this approach as slicing the shape into thin pieces. To find the total area, the process is: decide along which axis you want to slice (say slices perpendicular to x) find the size of a generic slice, as a function of the position x write out the sum that represents to the total you want transform the sum into an integral evaluate the integral
7 Integrals as Sums of Slices - 2 Example: Find the area between x = y 2 and y = x using horizontal slices
8 Integrals as Sums of Slices
9 Pyramid Volume - 1 Pyramid Volume A pyramid with its base being an equilateral triangle with sides 3 units long, is 5 units high. What is its volume? Helpful fact: the area of an equilateral 3 triangle with all sides length a is 4 a2 square units.
10 Pyramid Volume - 2
11 Cone Volume - 1 Cone Volume Example: Use a similar slicing strategy to find the volume of a cone of height h and bottom radius r.
12 Cone Volume - 2
13 Volumes of Revolution - Introduction - 1 Volumes of Revolution - Introduction Reading: Section 8.2 Wecanextendtheconeexampletofindthevolumeofmorecomplex spun shapes. These are often called volumes of revolution. Example: Consider the graph of y = e x shown below, and the solid we would build if we spun this shape around the x axis, cutting it off at x = 0 and x = What is the shape of any cut made perpendicular to the x axis?
14 Volumes of Revolution - Introduction - 2 Express the volume of a cut, x thick, in terms of the location of the cut, x Write down an integral that represents the total volume of the shape.
15 Evaluate the integral for the volume of the shape. Volumes of Revolution - Introduction - 3
16 Volumes of Revolution - Changing Rotation Axis - 1 Volumes of Revolution - Changing Rotation Axis Example: Now consider the shape we would get if we spun y = e x around the y axis. Suppose we cut the shape off between y = 0.1 and y = What is the shape of any cut made perpendicular to the x axis? Would this make a good way to cut up the shape? What is the shape of any cut made perpendicular to the y axis? Would this make a good way to cut up the shape?
17 Volumes of Revolution - Changing Rotation Axis - 2 Express the volume of a cut, y thick, in terms of the location of the cut, y Write down an integral that represents the total volume of the shape.
18 Evaluate the integral from the previous page. Volumes of Revolution - Changing Rotation Axis - 3
19 Volumes of Revolution - Changing Rotation Axis - 4
20 Volumes of Revolution - Slices as Rings - 1 Volumes of Revolution - Slices As Rings Example: The region bounded by y = 4x 2 x and y = 0 is rotated around the y-axis. Find the volume.
21 Volumes of Revolution - Slices as Rings - 2
22 Volumes of Revolution - Displaced Rotation Axis 1-1 Volumes of Revolution - Displaced Rotation Axis 1 Example: The region bounded by y = x 2, y = 0 and x = 2 is rotated around the line y = 3. Find the volume.
23 Volumes of Revolution - Displaced Rotation Axis 1-2
24 Volumes of Revolution - Displaced Rotation Axis 2-1 Volumes of Revolution - Displaced Rotation Axis 2 Example: The region bounded by y = x, y = 0 and x = 4 is rotated around the line x = 6. Find the volume.
25 Volumes of Revolution - Displaced Rotation Axis 2-2
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