Textbook Question
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«βΒΉ (πΒ³ β 2π) dπ = β3/4
(a) β«βΒΉ (4πβ2πΒ³) dπ
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.3.20a
Area functions for linear functions Consider the following functions Ζ and real numbers a (see figure).
(a) Find and graph the area function A (π) = β«βΛ£ Ζ(t) dt .
Ζ(t) = 2t + 5 , a = 0
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the area function A(x) = β«βΛ£ Ζ(t) dt, where Ζ(t) = 2t + 5 and a = 0. This represents the area under the curve of Ζ(t) from t = a to t = x.
Step 2: Set up the integral. Substitute the given function Ζ(t) = 2t + 5 into the integral: A(x) = β«βΛ£ (2t + 5) dt.
Step 3: Break the integral into parts. Use the linearity of integration to separate the terms: A(x) = β«βΛ£ 2t dt + β«βΛ£ 5 dt.
Step 4: Compute each integral. For β«βΛ£ 2t dt, use the power rule for integration: β« t^n dt = (t^(n+1))/(n+1). For β«βΛ£ 5 dt, treat 5 as a constant and integrate: β« c dt = c * t.
Step 5: Combine the results and simplify. After evaluating the definite integrals, combine the terms to express A(x) as a function of x. This will give the area function A(x).
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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 denoted as β«βΛ£ f(t) dt, where 'a' is the lower limit and 'x' is the upper limit. This concept is fundamental in calculating the area function A(x), which accumulates the area under the function f(t) from 'a' to 'x'.
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Definition of the Definite Integral
Area Function
The area function A(x) is defined as the integral of a function f(t) from a fixed point 'a' to a variable point 'x'. It quantifies the total area under the curve of f(t) from 'a' to 'x'. In this case, with f(t) = 2t + 5, A(x) will yield a linear function representing the area as 'x' changes.
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Graphing Area Functions
Graphing the area function A(x) involves plotting the accumulated area under the curve of f(t) as 'x' varies. The resulting graph typically shows how the area increases with 'x', reflecting the behavior of the original function. For linear functions like f(t) = 2t + 5, the area function will also be linear, making it easier to visualize the relationship between the two.
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Related Practice
Textbook Question
Working with area functions Consider the function Ζ and the points a, b, and c.
(a) Find the area function A (π) = β«βΛ£ Ζ(t) dt using the Fundamental Theorem.
Ζ(π) = cos π ; a = 0 , b = Ο/2 , c = Ο
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Textbook Question
Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.
(a) Find the mass of the left half of the rod (0 β€ x β€ 5) .
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Textbook Question
Approximating areas Estimate the area of the region bounded by the graph of Ζ(π) = xΒ² + 2 and the x-axis on [0, 2] in the following ways.
(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.
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Textbook Question
Use Table 5.6 to evaluate the following indefinite integrals.
(a) β« eΒΉβ°Λ£ dπ
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Textbook Question
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«βΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(a) A(2)
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