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 5.3.13d

Chapter 5, Problem 5.3.13d

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(d) F(4)

Verified step by step guidance

Step 1: Understand the problem. We are tasked with evaluating F(4), where F(x) = ∫₄ˣ ƒ(t) dt. This represents the net area under the curve of ƒ(t) from t = 4 to t = x. Specifically, F(4) means evaluating the integral from t = 4 to t = 4.

Step 2: Recall a key property of definite integrals. When the upper and lower limits of integration are the same, the integral evaluates to 0. Mathematically, ∫ₐₐ ƒ(t) dt = 0 for any function ƒ(t).

Step 3: Apply this property to the given integral. Since F(4) = ∫₄⁴ ƒ(t) dt, the integral evaluates to 0 because the interval of integration has no width.

Step 4: Confirm the reasoning using the graph. The graph shows areas labeled for different intervals, but since the interval from t = 4 to t = 4 has no length, no area is enclosed.

Step 5: Conclude that F(4) = 0 based on the mathematical property of definite integrals and the interpretation of the graph.

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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 between two points on the x-axis. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. In this context, the area functions A(x) and F(x) are defined as definite integrals, allowing us to evaluate the total area under the curve of the function f(t) from a specified lower limit to an upper limit.

Area Function

An area function, such as A(x) or F(x), is a function that gives the accumulated area under a curve from a specific starting point to a variable endpoint x. For example, A(x) = ∫₋₂ˣ f(t) dt calculates the area from -2 to x, while F(x) = ∫₄ˣ f(t) dt calculates the area from 4 to x. These functions help in understanding how the area changes as x varies.

Evaluating Area Functions

To evaluate an area function at a specific point, such as F(4), one must compute the definite integral from the lower limit to the specified upper limit. In this case, F(4) = ∫₄⁴ f(t) dt, which represents the area under the curve from 4 to 4. Since the limits are the same, the result is zero, indicating no area is accumulated over that interval.

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Related Practice

Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.

(e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)

Textbook Question

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

f(x) = x + 1 on [0,4]; n = 4

(d) Calculate the left and right Riemann sums.

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Textbook Question

Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.

(d) ∫₄⁶ (g(𝓍) ― f(𝓍) d𝓍

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

{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.

∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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Textbook Question

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

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

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.(d) F(4)

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Key Concepts

Definite Integral

Area Function

Evaluating Area Functions

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(d) F(4)