540
Chapter 5 | Integration
Theorem 5.2: Comparison Theorem i. If f ( x ) ≥0 for a ≤ x ≤ b , then
b
∫ a
f ( x ) dx ≥0.
ii. If f ( x ) ≥ g ( x ) for a ≤ x ≤ b , then
b
b
∫ a f ( x ) dx ≥ ∫ g ( x ) dx . iii. If m and M are constants such that m ≤ f ( x ) ≤ M for a ≤ x ≤ b , then m ( b − a ) ≤ ∫ a b f ( x ) dx ≤ M ( b − a ). a
Example 5.13 Comparing Two Functions over a Given Interval
Compare f ( x ) = 1+ x 2 and g ( x ) = 1+ x over the interval [0, 1].
Solution Graphing these functions is necessary to understand how they compare over the interval [0, 1]. Initially, when graphed on a graphing calculator, f ( x ) appears to be above g ( x ) everywhere. However, on the interval [0, 1], the graphs appear to be on top of each other. We need to zoom in to see that, on the interval [0, 1], g ( x ) is above f ( x ). The two functions intersect at x =0 and x =1 ( Figure 5.23 ).
Figure 5.23 (a) The function f ( x ) appears above the function g ( x ) except over the interval [0, 1] (b) Viewing the same graph with a greater zoom shows this more clearly.
We can see from the graph that over the interval [0, 1], g ( x ) ≥ f ( x ). Comparing the integrals over the specified interval [0, 1], we also see that ∫ 0 1 g ( x ) dx ≥ ∫ 0 1 f ( x ) dx ( Figure 5.24 ). The thin, red-shaded area shows just how much difference there is between these two integrals over the interval [0, 1].
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