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Chapter 6 | Applications of Integration
The technique we have just described is called the slicing method . To apply it, we use the following strategy.
Problem-Solving Strategy: Finding Volumes by the Slicing Method 1. Examine the solid and determine the shape of a cross-section of the solid. It is often helpful to draw a picture if one is not provided. 2. Determine a formula for the area of the cross-section. 3. Integrate the area formula over the appropriate interval to get the volume. Recall that in this section, we assume the slices are perpendicular to the x -axis. Therefore, the area formula is in terms of x and the limits of integration lie on the x -axis. However, the problem-solving strategy shown here is valid regardless of how we choose to slice the solid. Example 6.6 Deriving the Formula for the Volume of a Pyramid
We know from geometry that the formula for the volume of a pyramid is V = 1 3
Ah . If the pyramid has a square
a 2 h , where a denotes the length of one side of the base. We are going to use the
base, this becomes V = 1 3
slicing method to derive this formula.
Solution We want to apply the slicing method to a pyramid with a square base. To set up the integral, consider the pyramid shown in Figure 6.14 , oriented along the x -axis.
Figure 6.14 (a) A pyramid with a square base is oriented along the x -axis. (b) A two-dimensional view of the pyramid is seen from the side.
We first want to determine the shape of a cross-section of the pyramid. We know the base is a square, so the cross-sections are squares as well (step 1). Now we want to determine a formula for the area of one of these cross- sectional squares. Looking at Figure 6.14 (b), and using a proportion, since these are similar triangles, we have s a = x h or s = ax h . Therefore, the area of one of the cross-sectional squares is
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