Explanation Given: The equations of curve are y = 2 − x 2 and y = x . Calculation: Show the equations as below: y = 2 − x 2 (1) y = x (2) Plot a graph for the equations y = 2 − x 2 and y = x using the calculation as follows: Calculate y value using Equation (1) Substitute 0 for x in Equation (1). y = 2 − 0 2 = 2 Hence, the co-ordinate of ( x , y ) is ( 0 , 2 ) . Calculate y value using Equation (1) Substitute 1 for x in Equation (1). y = 2 − 1 2 = 1 Hence, the co-ordinate of ( x , y ) is ( 1 , 1 ) . Calculate y value using Equation (2) Substitute 0 for x in Equation (2). y = 0 The co-ordinate of ( x , y ) is ( 0 , 0 ) . Calculate y value using Equation (2). Substitute 1 for x in Equation (2). y = 1 The co-ordinate of ( x , y ) is ( 1 , 1 ) . Similarly calculate the coordinate values up to bound the region in the graph. Draw the region as in Figure 1. Calculate the area of the region: A = ∫ a b [ f ( x ) − g ( x ) ] d x (3) Calculate the x values: Substitute x for y in Equation (1). x = 2 − x 2 x 2 + x − 2 = 0 x 2 − x + 2 x − 2 = 0 x ( x − 1 ) + 2 ( x − 1 ) = 0 ( x − 1 ) ( x + 2 ) = 0 x 1 = 1 x 2 = − 2 Substitute ( − 2 ) for a , 1 for b , 2 − x 2 for [ f ( x ) ] , and x for [ g ( x ) ] in Equation (3). A = ∫ − 2 1 ( 2 − x 2 − x ) d x = ∫ − 2 1 ( − x 2 − x + 2 ) d x (4) Integrate Equation (4). A = [ − x 2 + 1 2 + 1 − x 1 + 1 1 + 1 + 2 x ] − 2 1 = [ − 1 3 x 3 − 1 2 x 2 + 2 x ] − 2 1 = [ − 1 3 ( 1 ) 3 − 1 2 ( 1 ) 2 + 2 ( 1 ) ] − [ − 1 3 ( − 2 ) 3 − 1 2 ( − 2 ) 2 + 2 ( − 2 ) ] = [ ( − 1 3 − 1 2 + 2 ) − ( 8 3 − 2 − 4 ) ] = 7 6 − ( − 10 3 ) = 27 6 = 9 2 Calculate the ( x ¯ ) coordinate of centroid: x ¯ = 1 A ∫ a b x [ f ( x ) − g ( x ) ] d x (5) Substitute ( − 2 ) for a , 1 for b , 9 2 for A , 2 − x 2 for [ f ( x ) ] , and x for [ g ( x ) ] in Equation (5). x ¯ = 1 9 2 ∫ − 2 1 x ( 2 − x 2 − x ) d x = 2 9 ∫ − 2 1 x ( − x 2 − x + 2 ) d x = 2 9 ∫ − 2 1 ( − x 3 − x 2 + 2 x ) d x (6) Integrate Equation (6)

8th Edition

ISBN: 9781305266636

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 8.3, Problem 30E

To determine

To find: The centroid (x¯,y¯) of the region bounded by the given curve.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.3, Problem 29E

Chapter 8.3, Problem 31E

Students have asked these similar questions

Sketch the region enclosed by the given curves. y = x2, 4 8 y = x2 + 4 y y y y 3 2 3 1 - 3 -2 -1 1 3 -3 -2 -3 -2 -1 1 2 3 -2 -1 1 2 -2 -1 -1 Find the area of the region.

Find the area of the region enclosed by the curve y=e-5x and the lines y=1 and x=2. -10 9 + e 10 10 -10 + 16 e-10 e-5 + 10 9 e-10 + 11 10

Find the area.

Chapter 8 Solutions

Single Variable Calculus

Ch. 8.1 - Use the arc length formula (3) to find the length...Ch. 8.1 - Use the arc length formula to find the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Find the exact length of the curve. 9. y = 1 +...Ch. 8.1 - Find the exact length of the curve. 10. 36y2 = (x2...

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.

The centroid ( x ¯ , y ¯ ) of the region bounded by the given curve.

Solution Summary: The author explains how to find the centroid (stackrel x), and calculate y value using Equation (1) Substitute 0 for X.

Concept explainers

Videos

To find: The centroid (x¯,y¯) of the region bounded by the given curve.

Want to see the full answer?

Chapter 8 Solutions