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The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.

If \( \int_0^1 f(x) \, dx = 4 \), what is the value of

\[ \int_0^1 2 f(x) \, dx ? \]

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If \( \int_0^1 f(x) \, dx = 2 \) and \( \int_0^1 g(x) \, dx = 6 \), what is the value of

\[ \int_0^1 \left( 6 f(x) + 2 g(x) \right) \, dx ? \]

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If \( \int_0 ^ {4} f(x) \, dx = 3 \), what is the value of

\[ \int_0^{4} \left( 3 - f(x) \right) \, dx ? \]

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If \( \int_0^{10} f(x) \, dx = 27 \) and \( \int_0^5 f(x) \, dx = 7 \), then what is the value of

\[ \int_5^{10} f(x) \, dx ? \]

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Suppose \(f(x)\) is an odd function and \(g(x)\) is an even function such that

\[ \int_0 ^ {5} f(x) \, dx = 7 \hspace{.6cm} \int_{5}^{15} f(x) \, dx = 8 \\ \int_{-5}^{0} g(x) \, dx = 1 \hspace{.6cm} \int_{5}^{15} g(x) \, dx = 2 \]

What is the value of \[ \int_{-5}^{15} \left( f(x) + g(x) \right) \, dx ? \]

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