Explain why ∫ₐᵇ ƒ ′(𝓍) d𝓍 = ƒ(b) ― ƒ(a)

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. 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).

Key Concepts

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.

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.

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.