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
1:46 minutesProblem 5.3.10
Textbook Question
Explain why ∫ₐᵇ ƒ ′(𝓍) d𝓍 = ƒ(b) ― ƒ(a)
Verified step by step guidance1
Step 1: Recall the Fundamental Theorem of Calculus, Part 1, which states that if a function ƒ is continuous on [a, b] and differentiable on (a, b), then the integral of its derivative ƒ′(𝓍) over [a, b] is equal to the net change in ƒ(𝓍) over that interval.
Step 2: Write the mathematical expression for the Fundamental Theorem of Calculus: ∫ₐᵇ ƒ′(𝓍) d𝓍 = ƒ(b) ― ƒ(a). This equation shows that the definite integral of the derivative of ƒ(𝓍) from a to b gives the difference between the values of ƒ(𝓍) at the endpoints b and a.
Step 3: Conceptually, the derivative ƒ′(𝓍) represents the rate of change of the function ƒ(𝓍). Integrating ƒ′(𝓍) over the interval [a, b] accumulates all the infinitesimal changes in ƒ(𝓍) over that interval, resulting in the total change in ƒ(𝓍) from a to b.
Step 4: To understand this geometrically, think of the integral ∫ₐᵇ ƒ′(𝓍) d𝓍 as calculating the area under the curve of ƒ′(𝓍) over [a, b]. This area corresponds to the net change in the original function ƒ(𝓍) over the interval.
Step 5: Finally, note that this relationship holds because differentiation and integration are inverse operations. The integral essentially 'undoes' the differentiation, leaving you with the original function evaluated at the endpoints: ƒ(b) ― ƒ(a).
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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 [a, b] and F is an antiderivative of f on that interval, then the integral of f from a to b equals F(b) - F(a). This theorem provides a powerful tool for evaluating definite integrals and establishes the relationship between the two main branches of calculus.
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Antiderivative
An antiderivative of a function f is another function F such that F' = f. In the context of the integral ∫ₐᵇ ƒ ′(𝓍) d𝓍, the function F is the antiderivative of f', meaning that F is the original function f before differentiation. This concept is crucial for understanding how integration reverses the process of differentiation.
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Definite Integral
A definite integral, represented as ∫ₐᵇ f(x) dx, calculates the net area under the curve of the function f(x) from x = a to x = b. It provides a numerical value that represents the accumulation of quantities, such as area, over an interval. The result of a definite integral is a specific number, which in the case of ∫ₐᵇ ƒ ′(𝓍) d𝓍, corresponds to the difference in the values of the antiderivative at the endpoints b and a.
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