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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.3.14d
Chapter 5, Problem 5.3.14d

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(d) F(8)
Graph of a function with labeled areas: green area = 8, red area = 5, red area = 11, showing integral calculations.

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Step 1: Understand the problem. We are tasked with evaluating F(8), where F(x) = ∫₂ˣ ƒ(t) dt. This represents the net area under the curve of ƒ(t) from t = 2 to t = 8.
Step 2: Analyze the graph. The graph shows the function ƒ(t) with labeled areas. The areas are given as follows: from t = 1 to t = 2, the area is 8 (positive, above the x-axis); from t = 2 to t = 3, the area is 5 (negative, below the x-axis); from t = 3 to t = 5, the area is 5 (positive, above the x-axis); and from t = 5 to t = 8, the area is 11 (negative, below the x-axis).
Step 3: Focus on the interval [2, 8]. Since F(8) involves the integral from t = 2 to t = 8, we only consider the areas between these bounds. Specifically, the areas are: from t = 2 to t = 3 (area = -5), from t = 3 to t = 5 (area = +5), and from t = 5 to t = 8 (area = -11).
Step 4: Compute the net area. Add the signed areas from t = 2 to t = 8: (-5) + (+5) + (-11). This sum represents the value of F(8).
Step 5: Conclude the process. The result of the sum in Step 4 gives the value of F(8). Ensure that the signs of the areas are correctly applied based on whether the region is above or below the x-axis.

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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 integral symbol and provides a numerical value that corresponds to the total area, accounting for areas above and below the x-axis. In the context of area functions, it helps determine the accumulated area from a starting point to a variable endpoint.
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Definition of the Definite Integral

Area Function

An area function, such as A(x) or F(x), is defined as the integral of a function from a fixed point to a variable upper limit. It quantifies the area under the curve of the function from the lower limit to x. This concept is crucial for evaluating specific values of the area function, as it allows us to compute the total area accumulated up to a certain point.
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Finding Area When Bounds Are Not Given

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval, then the integral of f from a to b is equal to F(b) - F(a). This theorem is essential for evaluating area functions, as it allows us to compute definite integrals by finding the difference of the antiderivative values at the endpoints.
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Fundamental Theorem of Calculus Part 1
Related Practice
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.


∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

Textbook Question

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

ƒ(𝓍) = x² ─ 1 on [2,4]; n = 4

(d) Calculate the left and right Riemann sums. 

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(d) If ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ is a constant function. 

Textbook Question

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

(a) ∫₀³ 5ƒ(𝓍) d𝓍

Textbook Question

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

{Use of Tech} ƒ(𝓍) = cos 𝓍 on [0. π/2]; n = 4

(d) Calculate the left and right Riemann sums.

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Textbook Question

Sigma notation Evaluate the following expressions.

(d)     5                                                                                                                                                                              

       ∑ (1 + n²)                                                                                                                                                                          

       n=1