If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.

d. ∫⁵₋₂ (-πg(x)) dx

검증된 단계별 안내

Recall the property of definite integrals that allows constants to be factored out: for any constant \(c\), \(\int_a^b c \cdot h(x) \, dx = c \int_a^b h(x) \, dx\).

Apply this property to the integral \(\int_{-2}^5 (-\pi g(x)) \, dx\), rewriting it as \(-\pi \int_{-2}^5 g(x) \, dx\).

Use the given value \(\int_{-2}^5 g(x) \, dx = 2\) and substitute it into the expression to get \(-\pi \times 2\).

Express the integral in terms of \(\pi\) and the known value without calculating the numerical product, as per instructions.

Thus, the value of \(\int_{-2}^5 (-\pi g(x)) \, dx\) is \(-2\pi\) times the integral of \(g(x)\) over \([-2,5]\), which is \(-2\pi\).

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주요 개념

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Properties of Definite Integrals

Definite integrals have linearity properties, meaning the integral of a sum is the sum of the integrals, and constants can be factored out. For example, ∫[a to b] c·f(x) dx = c·∫[a to b] f(x) dx. This allows simplification of integrals involving constants or sums of functions.

Given Integral Values and Interval Consistency

When solving problems involving definite integrals, it is crucial to pay attention to the integration limits. The given values correspond to specific intervals, and the integral's value depends on these limits. Ensuring the limits match the problem's integral is essential for correct evaluation.

Integral of a Scalar Multiple of a Function

If a function g(x) is multiplied by a scalar constant k, the integral over an interval is k times the integral of g(x) over that interval. Formally, ∫[a to b] k·g(x) dx = k·∫[a to b] g(x) dx. This property simplifies calculations involving constants like -π.

If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.d. ∫⁵₋₂ (-πg(x)) dx