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If
int_(3)^(14)f(x)dx=68 and
int_(11)^(14)f(x)dx=10, calculate
int_(3)^(11)f(x)dx

If $\int_{3}^{14} f(x) d x=68$ and $\int_{11}^{14} f(x) d x=10$ , calculate $\int_{3}^{11} f(x) d x$

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Evaluate definite integrals using the chain rule

Full solution

Q. If

\int_{3}^{14} f(x) d x=68

and

\int_{11}^{14} f(x) d x=10

, calculate

\int_{3}^{11} f(x) d x

Property of Integrals: We know $\int_{3}^{14}f(x)\,dx = 68$ and $\int_{11}^{14}f(x)\,dx = 10$ . Using the property of integrals, $\int_{a}^{c}f(x)\,dx = \int_{a}^{b}f(x)\,dx + \int_{b}^{c}f(x)\,dx$ .
Find $\int_{3}^{11}f(x)\,dx$ : Let's find $\int_{3}^{11}f(x)\,dx$ . We can write $\int_{3}^{14}f(x)\,dx$ as $\int_{3}^{11}f(x)\,dx + \int_{11}^{14}f(x)\,dx$ .
Equation Setup: So, $68 = \int_{3}^{11}f(x)\,dx + 10$ .
Isolate $\int_{3}^{11}f(x)\,dx$ : Subtract $10$ from both sides to isolate $\int_{3}^{11}f(x)\,dx$ . $\newline$ $\int_{3}^{11}f(x)\,dx = 68 - 10$ .
Calculate Difference: Calculate the difference: $68 - 10 = 58$ . $\newline$ $\int_{3}^{11}f(x)\,dx = 58$ .

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