Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.

(b) ∫₁⁶ (f(𝓍) ― g(𝓍)) d𝓍

검증된 단계별 안내

Step 1: Recall the property of integrals that states ∫ₐᵇ (ƒ(𝓍) ± g(𝓍)) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍 ± ∫ₐᵇ g(𝓍) d𝓍. This allows us to split the integral into two separate integrals.

Step 2: Apply the property to the given integral ∫₁⁶ (ƒ(𝓍) ― g(𝓍)) d𝓍. This becomes ∫₁⁶ ƒ(𝓍) d𝓍 ― ∫₁⁶ g(𝓍) d𝓍.

Step 3: Substitute the given values for the integrals: ∫₁⁶ ƒ(𝓍) d𝓍 = 10 and ∫₁⁶ g(𝓍) d𝓍 = 5.

Step 4: Perform the subtraction operation symbolically: 10 ― 5.

Step 5: The result of the subtraction gives the value of the integral ∫₁⁶ (ƒ(𝓍) ― g(𝓍)) d𝓍. This completes the evaluation process.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Properties of Definite Integrals

Definite integrals have several key properties, including linearity, which states that the integral of a sum of functions is the sum of their integrals. This means that for any two functions f and g, ∫(f + g) dx = ∫f dx + ∫g dx. Additionally, the integral of a constant multiplied by a function can be factored out: ∫k * f dx = k * ∫f dx. Understanding these properties is essential for evaluating integrals involving combinations of functions.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫ₐᵇ f(x) dx = F(b) - F(a). This theorem allows us to evaluate definite integrals by finding the antiderivative of the integrand. It is crucial for solving problems involving definite integrals, as it provides a method to compute the area under a curve.

Substitution in Integrals

Substitution is a technique used in integration to simplify the process of finding integrals. It involves changing the variable of integration to make the integral easier to evaluate. For example, if we let u = g(x), then the integral ∫f(g(x))g'(x)dx can be transformed into ∫f(u)du. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in a simpler form.

Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.(b) ∫₁⁶ (f(𝓍) ― g(𝓍)) d𝓍