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6.2 Notes
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6.2: Volumes by Disks and Washers
In this lesson, we wish to find ways to determine the volume of figures. Consider the following figure. The solid S can be sliced by parallel planes at each value of x, and these areas A(x) can be calculated.
Next, consider dividing the solid into "slabs" of equal width Δx. We can then choose sample points in each interval and use these to calculate the volume of each slab Si.
Finally, we add up the total volumes of these slabs to approximate the volume of the figure, and since this value gets more and more refined as the number of slabs gets larger, we will define the volume of S as the limit of these total volumes.
6.2 Notes
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Definition: If S is a solid that lies between x = a and x = b, then the volume of S is , where A(x) is the cross sectional area at any point x.
Example: Find the volume of a square pyramid with height h and side length s.
Example: Find the volume of the solid obtained by rotating the region bounded by y = , y = 0, and x = 1 around the xaxis.
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Theorem (The Disk Method): Let R be a region bounded by y = f(x), wheref(x) ≥ 0 on [a, b], y = 0, x = a, and x = b. The volume of the solid generated by revolving R around the xaxis is
Similar theorems can be made for rotating around the yaxis or vertical or horizontal lines that are not one of the axes. We will demonstrate these by example, but the basic idea is to remember that we are always integrating πr2.
Example: Find the volume of the solid obtained by rotating the region bounded byy = ex, y = 0, x = 0, and x = 1 around the xaxis.
6.2 Notes
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*Example: Find the volume of the solid obtained by rotating the region bounded byy = x2 + 2x and y = 0 around the xaxis.
Example: Find the volume of the solid obtained by rotating the region bounded byx = y, x = 0, and y = 2 around the yaxis.
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*Example: Find the volume of the solid obtained by rotating the region bounded byx = 1 y2, x = 0, and y = 0 around the yaxis.
Example: Find the volume of the solid obtained by rotating the region bounded byy = x and y = x2 around the xaxis.
6.2 Notes
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Theorem (The Washer Method): Let R be a region bounded by y = f(x), y = g(x) where f(x) ≥ g(x) on [a, b], x = a, and x = b. The volume of the solid generated by revolving R around the xaxis is
Similar theorems can be made for rotating around the yaxis or vertical or horizontal lines that are not one of the axes. We will demonstrate these by example, but the basic idea is to remember that we are always integrating πR2 πr2.
*Example: Find the volume of the solid obtained by rotating the region bounded byy = x and y = x2 around the yaxis.
6.2 Notes
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*Example: Find the volume of the solid obtained by rotating the region bounded byy = x and y = x2 around the yaxis.
Example: Find the volume of the solid obtained by rotating the region bounded byy = ex, y = 0, x = 0, and x = 1 around the line y = 3.
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*Example: Find the volume of the solid obtained by rotating the region bounded byx = 1 y2, x = 0, and y = 0 around the line x = 1.
*Example: Find the volume of the solid obtained by rotating the region bounded byy = ln x, x = 0, y = 1, and y = 2 around the line x = 2.
6.2 Notes
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*Example: Find the volume of the solid obtained by rotating the region bounded byy = x and y = x2 around the line x = 2.
*Example: Find the volume of the solid obtained by rotating the region bounded byy = x and y = x2 around the line y = 3.
