Evaluate ∫₃⁸ ƒ ′(t) dt , where ƒ ′ is continuous on [3, 8], ƒ(3) = 4, and ƒ(8) = 20 .

검증된 단계별 안내

Recognize that the integral ∫₃⁸ ƒ ′(t) dt represents the net change of the function ƒ(t) over the interval [3, 8]. This is based on the Fundamental Theorem of Calculus, which states that ∫ₐᵇ ƒ ′(x) dx = ƒ(b) - ƒ(a).

Identify the given values: ƒ(3) = 4 and ƒ(8) = 20. These represent the values of the function ƒ(t) at the endpoints of the interval [3, 8].

Apply the Fundamental Theorem of Calculus: Substitute the values of ƒ(8) and ƒ(3) into the formula ƒ(b) - ƒ(a). Specifically, calculate ƒ(8) - ƒ(3).

Set up the subtraction: ƒ(8) - ƒ(3) = 20 - 4. This represents the net change of the function ƒ(t) over the interval [3, 8].

Conclude that the integral ∫₃⁸ ƒ ′(t) dt is equal to the result of the subtraction performed in the previous step, which represents the total change in ƒ(t) over the interval.

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주요 개념

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

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is continuous on an interval, the integral of its derivative over that interval equals the difference in the function's values at the endpoints. Specifically, ∫ₐᵇ f'(t) dt = f(b) - f(a). This theorem is essential for evaluating definite integrals involving derivatives.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated as the limit of Riemann sums and provides a numerical value that reflects the accumulation of quantities, such as area or total change, between the two bounds. In this case, it helps determine the total change in the function f from t=3 to t=8.

Continuous Function

A continuous function is one that does not have any breaks, jumps, or holes in its graph over a given interval. For the Fundamental Theorem of Calculus to apply, the derivative f'(t) must be continuous on the interval [3, 8]. This ensures that the integral can be evaluated reliably and that the function f is well-defined at the endpoints.