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
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
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«ββΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(a) A (β2)
Textbook Question
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
β«ββ΄ 2βπ dπ
Textbook Question
Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(a) 1 + 2 + 3 + 4 + 5
Textbook Question
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«β^Ο/2 (cos ΞΈ β 2 sin ΞΈ) dΞΈ = β1
(a) β«β^Ο/2 (2 sin ΞΈ β cos ΞΈ) dΞΈ
Textbook Question
Approximating displacement The velocity in ft/s of an object moving along a line is given by v = 3tΒ² + 1 on the interval 0 β€ t β€ 4, where t is measured in seconds.
(a) Divide the interval [0,4] into n = 4 subintervals, [0,1] , [1.2] , [2,3] , and [3,4]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 4] (see part (a) of the figure)