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 5.1.5a

Chapter 5, Problem 5.1.5a

The velocity in ft/s of an object moving along a line is given by v = ƒ(t) on the interval 0 ≤ t ≤ 6 (see figure), where t is measured in seconds.

(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)

Verified step by step guidance

Divide the interval [0,6] into three subintervals: [0,2], [2,4], and [4,6].

For each subinterval, determine the velocity at the right endpoint by evaluating the graph of v = f(t). For [0,2], the right endpoint is t=2; for [2,4], the right endpoint is t=4; and for [4,6], the right endpoint is t=6.

Approximate the displacement on each subinterval by multiplying the velocity at the right endpoint by the length of the subinterval. For example, for [0,2], the displacement is v(2) * (2-0). Repeat this for [2,4] and [4,6].

Add the displacements from all three subintervals to estimate the total displacement of the object on [0,6].

Ensure that the units of the final displacement are consistent (e.g., ft) since velocity is given in ft/s and time in seconds.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Velocity Function

The velocity function, denoted as v = f(t), describes how the velocity of an object changes over time. In this context, it provides the speed of the object at any given moment t within the specified interval. Understanding this function is crucial for estimating displacement, as it directly influences how far the object travels during each subinterval.

Subintervals

Subintervals are smaller segments into which a larger interval is divided for analysis. In this problem, the interval [0, 6] is divided into three subintervals: [0, 2], [2, 4], and [4, 6]. This division allows for the approximation of displacement by evaluating the velocity at the right endpoint of each subinterval, simplifying the calculation of the total distance traveled.

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve, which in this case represents the displacement of the object. By using the right endpoint of each subinterval to determine the height of rectangles, the sum of the areas of these rectangles provides an estimate of the total displacement over the interval [0, 6]. This concept is fundamental in calculus for understanding integration and area calculations.

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Related Practice

Textbook Question

Working with area functions Consider the function ƒ and the points a, b, and c.

(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.

ƒ(𝓍) = cos 𝓍 ; a = 0 , b = π/2 , c = π

Textbook Question

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 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.

Textbook Question

Sigma notation Evaluate the following expressions.

(a) 10

∑ κ

κ=1

Textbook Question

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(e) ∫₋₂² 3𝓍ƒ(𝓍)d𝓍

Textbook Question

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(a) ∫₋₄⁴ ƒ(𝓍) d𝓍

Textbook Question

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.