Chapter 6 | Applications of Integration
627
Solution The region is depicted in the following figure.
Figure 6.5 This graph shows the region below the graph of f ( x ) and above the graph of g ( x ).
We first need to compute where the graphs of the functions intersect. Setting f ( x ) = g ( x ), we get f ( x ) = g ( x ) 9− ⎛ ⎝ x 2 ⎞ ⎠ 2 = 6− x
9− x x 36− x 2 = 24−4 x 2 4 = 6−
x 2 −4 x −12 = 0 ( x −6)( x +2) = 0. The graphs of the functions intersect when x =6 or x =−2, so we want to integrate from −2 to 6. Since f ( x ) ≥ g ( x ) for −2≤ x ≤6, we obtain A = ∫ ⎡ ⎣ f ( x )− g ( x ) ⎤ ⎦ dx
a b
6 ⎡ ⎣
⎤ ⎦ ⎥ dx = ∫
6 ⎡
x ⎤
⎛ ⎝ x 2
⎞ ⎠
2
2 4 +
⎢ 9−
= ∫
⎣ 3− x
⎦ dx
−(6− x )
−2
−2
⎤ ⎦ | −2 6
⎡ ⎣ 3 x − x
3 12 +
x 2 2
=
= 64 3 .
The area of the region is 64/3 units 2 .
6.2 If R is the region bounded above by the graph of the function f ( x ) = x and below by the graph of the function g ( x ) = x 4 , find the area of region R .
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