Question

Suppose \int_{-2}^{2}f(x)dx=4,\int_{2}^{5}f(x)dx=3, \int_{-2}^{5}g(x)dx=2. Is the following statement true? f(x)\leq g(x) on the interval -2\leq x\leq 5

Applications of integrals
ANSWERED
asked 2021-05-03
Suppose \(\int_{-2}^{2}f(x)dx=4,\int_{2}^{5}f(x)dx=3, \int_{-2}^{5}g(x)dx=2\).
Is the following statement true?
\(f(x)\leq g(x)\) on the interval \(-2\leq x\leq 5\)

Expert Answers (1)

2021-05-04
Step 1
Given:
\(\int_{-2}^{2}f(x)dx=4\)
\(\int_{2}^{5}f(x)dx=3\)
\(\int_{-2}^{5}g(x)dx=2\)
Step 2
Using properties of Definite Integrals:
\(\int_{a}^{b}[f(x)\pm g(x)]dx=\int_{a}^{b}f(x)dx\pm \int_{a}^{b}g(x)dx\)
\(\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx=\int_{a}^{b}f(x)dx\ \ \ [c\in(a,b)]\)
\(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\)
Step 3
Now,
\(\int_{-2}^{5}f(x)dx=\int_{-2}^{2}f(x)dx+\int_{2}^{5}f(x)dx=4+3=7\)
\(\int_{-2}^{5}g(x)dx=2\)
\(\therefore\) On interval \(-2\leq x\leq 5\)
\(f(x)\geq g(x)\)
False Statement
35
 
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