6.6 Properties of Definite Integrals

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Define the property of definite integrals that states the integral of a function over an interval can be expressed as the difference of integrals over subintervals.

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If \( a < b < c \), then \( \int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx \).

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What is the effect of reversing the limits of integration on a definite integral?

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Reversing the limits of integration changes the sign of the integral: \( \int_a^b f(x)dx = -\int_b^a f(x)dx \).

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-3

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10

Front

What is the relationship between the integral of a function and its constant multiple?

Back

For any constant \( k \), \( \int_a^b kf(x)dx = k\int_a^b f(x)dx \).

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What is the integral of a sum of functions?

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\( \int_a^b [f(x) + g(x)]dx = \int_a^b f(x)dx + \int_a^b g(x)dx \).

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34.

Front

What is the property of definite integrals regarding even and odd functions?

Back

For an even function \( f(x) \), \( \int_{-a}^a f(x)dx = 2\int_0^a f(x)dx \); for an odd function, \( \int_{-a}^a f(x)dx = 0 \).