Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.3.87a

Capitolo 5, Problema 5.3.87a

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.

Guida verificata passo dopo passo

Step 1: Understand the problem. We are tasked with matching the graph of the function f(t) (given in the first image) with the graph of its corresponding area function A(x) = ∫₀ˣ f(t) dt (given in the second image). The area function A(x) represents the accumulated area under the curve of f(t) from t = 0 to t = x.

Step 2: Analyze the graph of f(t). The graph of f(t) in image (a) is a constant function, where f(t) = c (a positive constant) for all t in the interval [0, b]. This means the value of f(t) does not change with t, and the area under the curve will grow linearly as x increases.

Step 3: Determine the behavior of A(x). Since f(t) is constant, the integral A(x) = ∫₀ˣ f(t) dt will result in a linear function. Specifically, A(x) will increase at a constant rate because the area under a constant function is proportional to the width of the interval.

Step 4: Match A(x) with the correct graph. Among the graphs labeled A–D in the second image, the graph labeled (C) shows a straight line, which corresponds to a linear function. This matches the behavior of A(x) derived from the constant function f(t).

Step 5: Conclude the match. The graph of f(t) in image (a) corresponds to the area function A(x) in graph (C). This is because the constant rate of change in f(t) leads to a linear accumulation of area in A(x).

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Definite Integral

A definite integral represents the signed area under a curve defined by a function f(t) from a lower limit to an upper limit. It is denoted as ∫ₐᵇ f(t) dt, where 'a' and 'b' are the bounds of integration. This concept is crucial for understanding how area functions A(x) are derived from the original function f(t) by integrating it over a specified interval.

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Definition of the Definite Integral

Area Function

An area function A(x) is defined as the integral of a function f(t) from a fixed point (usually 0) to a variable upper limit x. Mathematically, it is expressed as A(x) = ∫₀ˣ f(t) dt. This function provides a way to visualize how the area under the curve of f(t) accumulates as x changes, which is essential for matching the area functions with their corresponding original functions.

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Finding Area When Bounds Are Not Given

Graphical Interpretation of Integrals

The graphical interpretation of integrals involves visualizing the area under the curve of a function f(t) as the integral is computed. The shape and behavior of the area function A(x) can be analyzed by observing how the area accumulates as x increases. Understanding this relationship helps in matching the graphs of functions with their corresponding area functions, as the characteristics of the area function reflect the properties of the original function.

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Integration by Parts for Definite Integrals

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.

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