Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.R.15

Chapter 6, Problem 6.R.15

14–25. {Use of Tech} Areas of regions Determine the area of the given region.

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Identify the curves given: the upper curve is \(y = \sin 2x - 1\) and the lower curve is \(y = -\cos 2x\) over the interval where they intersect.

Find the points of intersection by setting \(\sin 2x - 1 = -\cos 2x\) and solving for \(x\). This will give the limits of integration.

Set up the integral for the area of the shaded region as the integral of the difference between the upper and lower curves: \(\int_{a}^{b} \left[(\sin 2x - 1) - (-\cos 2x)\right] \, dx\) where \(a\) and \(b\) are the intersection points.

Simplify the integrand to \(\int_{a}^{b} (\sin 2x - 1 + \cos 2x) \, dx\) to prepare for integration.

Integrate the function term-by-term with respect to \(x\), then evaluate the definite integral using the limits \(a\) and \(b\) to find the area.

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Key Concepts

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

Finding Points of Intersection

To determine the area between two curves, first find their points of intersection by setting the functions equal to each other. These points serve as the limits of integration, defining the interval over which the area is calculated.

Definite Integral for Area Between Curves

The area between two curves y = f(x) and y = g(x) over an interval [a, b] is found by integrating the difference f(x) - g(x) with respect to x. This integral sums the vertical distances between the curves across the interval.

Trigonometric Functions and Their Properties

Understanding the behavior of trigonometric functions like sin(2x) and cos(2x) is essential, including their periodicity and transformations. This knowledge helps in accurately setting up the integral and interpreting the graph.

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Textbook Question

14–25. {Use of Tech} Areas of regions Determine the area of the given region.

The region bounded by y = x²,y = 2x²−4x, and y = 0

Textbook Question

Comparing volumes Let R be the region bounded by y=1/x^p and the x-axis on the interval [1, a], where p>0 and a>1 (see figure). Let Vₓ and Vᵧ be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively.

d. Find a general expression for Vᵧ in terms of a and p. Note that p=2 is a special case. What is Vᵧ when p=2?

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Textbook Question

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:

b. Apply the shell method and integrate with respect to x.

Textbook Question

Position, displacement, and distance A projectile is launched vertically from the ground at t=0, and its velocity in flight (in m/s) is given by v(t)=20−10t. Find the position, displacement, and distance traveled after t seconds, for 0≤t≤4.

Textbook Question

Surface area and volume Let f(x) = 1/3 x³ and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2].

b. Find the volume of the solid generated when R is revolved about the y-axis.

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Textbook Question

An area function Consider the functions y = x²/a and y = √x/a, where a>0. Find A(a), the area of the region between the curves.