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Definite Integrals

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.

Basic Properties of Integrals

         

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

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

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

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

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

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

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

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

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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