Explanation Given information: The two curves has a function of y = cos x and y = x + 2 sin 4 x . Calculation: Procedure to sketch the region bounded by the two curves is explained below: Draw the graph for the function y = cos x by substituting different values for x . Similarly in the same graph plot for the function y = x + 2 sin 4 x by substituting different values for x . Shade the region lies between the intersecting points of the curves. The region enclosed by the curves y = cos x and y = x + 2 sin 4 x is shown in Figure 1. Refer to Figure 1. The curves intersects at x = − 1.9 , x = − 1.2 and x = 0.6 . The curves are bounded by the top and bottom curve. Hence, the integration can be done with respect to x . Find the area of the region bounded by the curves using the relation: A = ∫ a b ( f ( x ) − g ( x ) ) d x (1) Here, the top curve is f ( x ) , the bottom curve is g ( x ) , the lower limit is a , and the upper limit is b . Rearrange Equation (1) for the limits − 1.9 ≤ x ≤ − 1.2 and − 1.2 ≤ x ≤ 0.6 as shown below. A = ∫ − 1.9 − 1.2 ( f ( x ) − g ( x ) ) d x + ∫ − 1.2 0.6 ( g ( x ) − f ( x ) ) d x (2) Substitute x + 2 sin 4 x for f ( x ) and cos x for g ( x ) in Equation (1) A = ∫ − 1.9 − 1.2 ( ( x + 2 sin 4 x ) − cos x ) d x + ∫ − 1.2 0.6 ( cos x − ( x + 2 sin 4 x ) ) d x = ∫ − 1.9 − 1.2 ( x + 2 sin 4 x − cos x ) d x + ∫ − 1.2 0.6 ( cos x − x − 2 sin 4 x ) d x = A 1 + A 2 (3) Find the value of A 1 . A 1 = ∫ − 1.9 − 1.2 ( x + 2 sin 4 x − cos x ) d x (4) Show the reduction formula: ∫ sin n x d x = n − 1 n ∫ sin n − 2 x d x − cos x sin n − 1 x n Apply the reduction formula in Equation (4) with n = 4 . A 1 = ( x 2 2 + 2 ( 3 4 ∫ sin 2 x d x − cos x sin 3 x 4 ) − sin x ) − 1.9 − 1.2 (5) Apply the reduction formula in Equation (5) with n = 2 . A 1 = ( x 2 2 + ( 3 2 ( 1 2 ∫ d x − cos x sin x 2 ) − cos x sin 3 x 2 ) − sin x ) − 1.9 − 1.2 = ( x 2 2 + ( 3 4 ( x ) − 3 4 cos x sin x ) − cos x sin 3 x 2 − sin x ) − 1.9 − 1.2 = ( x 2 2 + 3 4 x − 1 2 cos x sin 3 x − 3 4 cos x sin x − sin x ) − 1.9 − 1.2 = 1 4 ( 2 x 2 + 3 x − 2 cos x sin 3 x − 3 cos x sin x − 4 sin x ) − 1

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Videos

Question

Chapter 6.1, Problem 44E

To determine

To draw: The region enclosed by the given curves.

The area of the region enclosed by the curves.

Expert Solution & Answer

Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 7E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To draw: The region enclosed by the given curves. The area of the region enclosed by the curves.

Solution Summary: The author illustrates the procedure to draw the region enclosed by the curves.

Concept explainers

Videos

To draw: The region enclosed by the given curves. The area of the region enclosed by the curves.

Chapter 6 Solutions

To draw: The region enclosed by the given curves.

The area of the region enclosed by the curves.