Chapter 6 | Applications of Integration
637
Figure 6.12 A solid with a varying cross-section.
We want to divide S into slices perpendicular to the x -axis. As we see later in the chapter, there may be times when we want to slice the solid in some other direction—say, with slices perpendicular to the y -axis. The decision of which way to slice the solid is very important. If we make the wrong choice, the computations can get quite messy. Later in the chapter, we examine some of these situations in detail and look at how to decide which way to slice the solid. For the purposes of this section, however, we use slices perpendicular to the x -axis. Because the cross-sectional area is not constant, we let A ( x ) represent the area of the cross-section at point x . Now let P = ⎧ ⎩ ⎨ x 0 , x 1 …, X n ⎫ ⎭ ⎬ be a regular partition of ⎡ ⎣ a , b ⎤ ⎦ , and for i =1, 2,… n , let S i represent the slice of S stretching from
x i −1 to x i . The following figure shows the sliced solid with n =3.
Figure 6.13 The solid S has been divided into three slices perpendicular to the x -axis.
Finally, for i =1, 2,… n , let x i * be an arbitrary point in [ x i −1 , x i ]. Then the volume of slice S i can be estimated by V ⎛ ⎝ S i ⎞ ⎠ ≈ A ⎛ ⎝ x i * ⎞ ⎠ Δ x . Adding these approximations together, we see the volume of the entire solid S can be approximated by V ( S ) ≈ ∑ i =1 n A ⎛ ⎝ x i * ⎞ ⎠ Δ x . By now, we can recognize this as a Riemann sum, and our next step is to take the limit as n →∞. Then we have
⎞ ⎠ Δ x = ∫ a b
→∞ ∑ i =1 n
A ⎛
V ( S ) = lim n
A ( x ) dx .
⎝ x i *
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