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.

a. ∫²₋₂ ƒ(x) dx

검증된 단계별 안내

Recall the linearity property of integrals: for any constant \( c \), \( \int_a^b c f(x) \, dx = c \int_a^b f(x) \, dx \).

Given \( \int_{-2}^2 3f(x) \, dx = 12 \), use the linearity property to express this as \( 3 \int_{-2}^2 f(x) \, dx = 12 \).

To find \( \int_{-2}^2 f(x) \, dx \), divide both sides of the equation by 3, resulting in \( \int_{-2}^2 f(x) \, dx = \frac{12}{3} \).

Simplify the right-hand side to express the integral \( \int_{-2}^2 f(x) \, dx \) in terms of a numerical value (do not calculate the final number here).

This gives you the value of \( \int_{-2}^2 f(x) \, dx \) based on the information provided.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Definite Integrals

Definite integrals have properties such as linearity, which allows constants to be factored out and integrals to be split or combined over intervals. For example, ∫_a^b c·f(x) dx = c·∫_a^b f(x) dx, and ∫_a^c f(x) dx + ∫_c^b f(x) dx = ∫_a^b f(x) dx.

Linearity of Integration

Integration is a linear operation, meaning the integral of a sum is the sum of the integrals, and constants can be factored out. This helps in manipulating given integrals to find unknown values by breaking down or combining integrals.

Using Given Integral Values to Find Unknowns

Given specific integral values over certain intervals, you can use properties of integrals to solve for unknown integrals over subintervals or related functions by setting up equations and applying algebraic manipulation.

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.a. ∫²₋₂ ƒ(x) dx