Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.3.87d

Chapter 5, Problem 5.3.87d

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.

Verified step by step guidance

Recall that the area function \(A(x) = \int_0^x f(t) \, dt\) represents the accumulated area under the curve of \(f(t)\) from 0 to \(x\). The derivative of \(A(x)\) is \(f(x)\), i.e., \(A'(x) = f(x)\).

Analyze the graph of \(f(t)\) (graph d): Identify where \(f(t)\) is positive, negative, increasing, or decreasing. Notice that \(f(t)\) starts at 0, goes negative, then positive, and returns to 0 at \(t = b\).

Look at each candidate graph for \(A(x)\) (graphs A, B, C, D) and consider the slope at each point, since the slope of \(A(x)\) at \(x\) must equal \(f(x)\) at that point.

Match the behavior of \(f(t)\) with the slope of each \(A(x)\) graph: For example, where \(f(t)\) is negative, \(A(x)\) should be decreasing; where \(f(t)\) is positive, \(A(x)\) should be increasing; and where \(f(t)\) crosses zero, \(A(x)\) should have a horizontal tangent (slope zero).

Use these observations to pair the graph of \(f(t)\) with the correct graph of \(A(x)\) by checking which \(A(x)\) graph's slope pattern matches the \(f(t)\) graph's values.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral as Area Function

The definite integral of a function f(t) from 0 to x, denoted A(x) = ∫₀ˣ f(t) dt, represents the net area between the graph of f and the t-axis over [0, x]. Positive areas above the axis add to A(x), while areas below subtract, affecting the shape of the area function.

Relationship Between a Function and Its Integral

The area function A(x) is an antiderivative of f(x), meaning A'(x) = f(x). This implies that the slope of the graph of A at any point x equals the value of f at x. Thus, where f is positive, A is increasing; where f is negative, A is decreasing.

Interpreting Graphs to Match Functions and Area Functions

Matching graphs of f and A requires analyzing where f is positive or negative and how the area accumulates. For example, if f dips below zero, A will decrease in that interval. Points where f crosses zero correspond to local maxima or minima in A, reflecting changes in slope direction.

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Related Practice

Textbook Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.

ƒ(𝓍) = 2x + 1 on [0,4] ; n = 4

d) Calculate the midpoint Riemann sum.

Textbook Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.

{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4

(d) Calculate the midpoint Riemann sum.

Textbook Question

Use Table 5.6 to evaluate the following definite integrals.

(d) ∫₀^π/¹⁶ sec ² 4𝓍 d𝓍

Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..

∫₀² (𝓍²―2) d𝓍 ; n = 4

Textbook Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(d) 1 + 1/2 + 1/3 + 1/4

Textbook Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.

ƒ(𝓍) = 1/x on [1,6] ; n = 5

(d) Calculate the midpoint Riemann sum.