Chapter 6 | Applications of Integration
631
This time, we are going to partition the interval on the y -axis and use horizontal rectangles to approximate the area between the functions. So, for i =0, 1, 2,…, n , let Q ={ y i } be a regular partition of ⎡ ⎣ c , d ⎤ ⎦ . Then, for i =1, 2,…, n , choose a point y i * ∈ [ y i −1 , y i ], then over each interval [ y i −1 , y i ] construct a rectangle that extends horizontally from v ⎛ ⎝ y i * ⎞ ⎠ ⎞ ⎠ . Figure 6.9 (a) shows the rectangles when y i * is selected to be the lower endpoint of the interval and n =10. Figure 6.9 (b) shows a representative rectangle in detail. to u ⎛ ⎝ y i *
Figure 6.9 (a) Approximating the area between the graphs of two functions, u ( y ) and v ( y ), with rectangles. (b) The area of a typical rectangle.
The height of each individual rectangle is Δ y and the width of each rectangle is u ⎛ ⎝ y i * ⎞ ⎠ − v ⎛ ⎝ y i * ⎞
⎠ . Therefore, the area
between the curves is approximately
A ≈ ∑ i =1 n
⎡ ⎣ u
⎤ ⎦ Δ y .
⎛ ⎝ y i *
⎞ ⎠ − v
⎛ ⎝ y i *
⎞ ⎠
This is a Riemann sum, so we take the limit as n →∞, obtaining
→∞ ∑ i =1 n
d
⎞ ⎠ ⎤ ⎦ Δ y = ∫
⎡ ⎣ u
⎛ ⎝ y i *
⎞ ⎠ − v
⎛ ⎝ y i *
⎡ ⎣ u ( y )− v ( y ) ⎤
A = lim n
⎦ dy .
c
These findings are summarized in the following theorem.
Theorem 6.3: Finding the Area between Two Curves, Integrating along the y -axis Let u ( y ) and v ( y ) be continuous functions such that u ( y ) ≥ v ( y ) for all y ∈ ⎡ ⎣ c , d ⎤ ⎦ . Let R denote the region bounded on the right by the graph of u ( y ), on the left by the graph of v ( y ), and above and below by the lines y = d and y = c , respectively. Then, the area of R is given by (6.2) A = ∫ c d ⎡ ⎣ u ( y )− v ( y ) ⎤ ⎦ dy .
Example 6.5 Integrating with Respect to y
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