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 5.3.87c

Chapter 5, Problem 5.3.87c

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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Observe the graph of the function f(t) in the first image. The graph is a parabola that starts at the origin, increases to a maximum point, and then decreases back to zero at t = b. This indicates that f(t) is positive over the interval [0, b], and the area under the curve will first increase and then decrease as x approaches b.

Recall that the area function A(x) = ∫₀ˣ f(t) dt represents the accumulated area under the curve of f(t) from t = 0 to t = x. Since f(t) is positive, A(x) will initially increase as x increases. However, as f(t) decreases after the maximum point, the rate of increase of A(x) will slow down, and A(x) will eventually stop increasing when x = b.

Compare the behavior of A(x) with the given graphs (A–D). The graph of A(x) that matches this behavior will start at 0, increase to a maximum value, and then decrease back to 0 at x = b. This matches the graph labeled (B).

Verify the match by considering the derivative relationship: A'(x) = f(x). The derivative of the area function A(x) should match the shape of f(t). Since the graph of A(x) in (B) has a slope that increases, reaches a maximum, and then decreases, it aligns with the shape of f(t) in the first image.

Conclude that the function f(t) in the first image corresponds to the area function A(x) in graph (B).

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Key Concepts

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

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 the area function A(x) accumulates the values of f(t) as x varies from 0 to x.

Area Function

The area function A(x) is defined as A(x) = ∫₀ˣ f(t) dt, which calculates the total area under the curve of f(t) from 0 to x. This function provides insight into how the area changes as x increases, and its graph typically reflects the accumulation of area, showing increasing or decreasing trends based on the behavior of f(t).

Relationship Between Functions and Their Area Functions

The relationship between a function f(t) and its area function A(x) is characterized by the Fundamental Theorem of Calculus, which states that the derivative of the area function A(x) is equal to the original function f(t). This means that the slope of the area function at any point x corresponds to the value of the function f(t) at that point, linking the two concepts through differentiation and integration.

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Textbook Question

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

Textbook Question

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

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

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Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

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Textbook Question

Working with area functions Consider the function ƒ and the points a, b, and c.

ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

Textbook Question

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.

(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

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Textbook Question

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

∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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