Textbook Question
Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.
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8. Definite Integrals
Fundamental Theorem of Calculus
1:31 minutesProblem 5.3.11
Textbook Question
Evaluate ∫₃⁸ ƒ ′(t) dt , where ƒ ′ is continuous on [3, 8], ƒ(3) = 4, and ƒ(8) = 20 .
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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