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Answer the following True or False. $\newline$ If $\int_{a}^{b} f(x) \, dx$ converges and $a < c < b$ , then $\int_{a}^{c} f(x) \, dx$ also converges. $\newline$ True $\newline$ False $\newline$

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Properties of matrices

Full solution

Q. Answer the following True or False.

\newline

If

\int_{a}^{b} f(x) \, dx

converges and

a < c < b

, then

\int_{a}^{c} f(x) \, dx

also converges.

\newline

True

\newline

False

\newline

Consider Integral Convergence: Let's consider the given integral from $a$ to $b$ of $f(x)$ . If this integral converges, it means that the area under the curve of $f(x)$ from $a$ to $b$ is finite.
Subset Interval Analysis: Now, let's consider the integral from $a$ to $c$ of $f(x)$ , where $a < c < b$ . Since $c$ is between $a$ and $b$ , the interval from $a$ to $c$ is a subset of the interval from $a$ to $b$ .
No Singularities or Divergence: The convergence of the integral from $a$ to $b$ of $f(x)$ implies that the function $f(x)$ does not have any singularities or behaviors that would cause the integral to diverge in the interval $[a, b]$ .
Convergence Implication: Since the interval from $a$ to $c$ is contained within the interval from $a$ to $b$ , and we know that there are no issues in the larger interval that would prevent convergence, the integral from $a$ to $c$ must also converge.
True Statement Validation: Therefore, the statement is true: if the integral from $a$ to $b$ of $f(x)$ converges, then the integral from $a$ to $c$ of $f(x)$ also converges for any $c$ such that $a < c < b$ .

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\newline

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x

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