Textbook Question
Area of regions Compute the area of the region bounded by the graph of Ζ and the π-axis on the given interval. You may find it useful to sketch the region.
Ζ(π) = 16βπΒ² on [β4, 4]
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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.R.96a
Velocity to displacement An object travels on the π-axis with a velocity given by v(t) = 2t + 5, for 0 β€ t β€ 4.
(a) How far does the object travel, for 0 β€ t β€ 4 ?
Verified step by step guidance1
Step 1: Recognize that the displacement of the object can be found by integrating the velocity function v(t) = 2t + 5 over the given time interval [0, 4]. The formula for displacement is: \( s(t) = \int v(t) \, dt \).
Step 2: Set up the definite integral for displacement: \( \int_{0}^{4} (2t + 5) \, dt \). This represents the total distance traveled by the object from t = 0 to t = 4.
Step 3: Break the integral into two parts for easier computation: \( \int_{0}^{4} 2t \, dt + \int_{0}^{4} 5 \, dt \).
Step 4: Compute each integral separately. For \( \int_{0}^{4} 2t \, dt \), use the power rule of integration: \( \int t^n \, dt = \frac{t^{n+1}}{n+1} \). For \( \int_{0}^{4} 5 \, dt \), treat 5 as a constant and multiply it by the length of the interval.
Step 5: Add the results of the two integrals together to find the total displacement. This will give the total distance traveled by the object over the interval [0, 4].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity
Velocity is the rate of change of an object's position with respect to time. In this context, the velocity function v(t) = 2t + 5 describes how the object's speed changes over time. Understanding velocity is crucial for determining how far the object travels over a given time interval.
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Derivatives Applied To Velocity
Displacement
Displacement refers to the change in position of an object and can be calculated as the integral of the velocity function over a specific time interval. In this case, to find the total distance traveled by the object from t = 0 to t = 4, we need to integrate the velocity function v(t) over that interval.
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10:17Using The Velocity Function
Definite Integral
A definite integral calculates the accumulation of quantities, such as area under a curve, over a specified interval. In this problem, we will use the definite integral of the velocity function from t = 0 to t = 4 to find the total distance traveled by the object, which is essential for solving the question.
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
Integration by Riemann sums Consider the integral β«ββ΄ (3πβ 2) dπ.
(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .
Textbook Question
(b) Find the average value of Ζ shown in the figure on the interval [2,6] and then find the point(s) c in (2, 6) guaranteed to exist by the Mean Value Theorem for Integrals.
Textbook Question
Properties of integrals Suppose β«ββ΄ Ζ(π) dπ = 6 , β«ββ΄ g(π) dπ = 4 and β«ββ΄ Ζ(π) dπ = 2 . Evaluate the following integrals or state that there is not enough information.
ββ«βΒΉ 2Ζ(π) dπ
Textbook Question
Evaluating integrals Evaluate the following integrals.
β«Ο/β^Ο/Β³ (secΒ² t + cscΒ² t) dt
Textbook Question
Function defined by an integral Let Ζ(π) = β«βΛ£ (t β 1)ΒΉβ΅ (tβ2)βΉ dt .
(c) For what values of π does Ζ have local minima? Local maxima?