Explanation Given: The equation of curve is y = x . The region lies between 0 and 4. Calculation: Show the equation as below: y = x (1) Plot a graph for the equation y = x and x = 4 using the calculation as follows: Calculate y value using Equation (1) Substitute 0 for x in Equation (1). y = 0 = 0 Hence, the co-ordinate of ( x , y ) is ( 0 , 0 ) . Calculate y value using Equation (1) Substitute 1 for x in Equation (1). y = 1 = ± 1 Hence, the co-ordinate of ( x , y ) is ( 1 , ± 1 ) . Calculate y value using Equation (1) Substitute 2 for x in Equation (1). y = 2 = ± 1.414 Hence, the co-ordinate of ( x , y ) is ( 2 , ± 1.414 ) . Calculate y value using Equation (1) Substitute 3 for x in Equation (1). y = 3 = ± 1.732 Hence, the co-ordinate of ( x , y ) is ( 3 , ± 1.732 ) . Calculate y value using Equation (1) Substitute 4 for x in Equation (1). y = 4 = ± 2 Hence, the co-ordinate of ( x , y ) is ( 4 , ± 2 ) . Draw the region as in Figure 1. Refer to Figure 1 The coordinate ( x ¯ ) of the region is 2.3 The coordinate ( y ¯ ) of the region is 0.8 Hence, the location ( x ¯ , y ¯ ) of the region bounded by the given curve is ( 2.3 , 0.8 ) _ . Calculate the area of the region: A = ∫ a b [ f ( x ) ] d x (2) Substitute 0 for a , 4 for b , and x for [ f ( x ) ] in Equation (2). A = ∫ 0 4 x d x = ∫ 0 4 x 1 2 d x (3) Integrate Equation (3). A = [ x 1 2 + 1 1 2 + 1 ] 0 4 = [ x 3 2 3 2 ] 0 4 = [ 2 x 3 2 3 ] 0 4 = [ 2 ( 4 ) 3 2 3 − 2 ( 0 ) 3 2 3 ] = 16 3 Calculate the ( x ¯ ) coordinate of centroid: x ¯ = 1 A ∫ a b x [ f ( x ) ] d x (4) Substitute 0 for a , 4 for b , 16 3 for A , and x for [ f ( x ) ] in Equation (4)

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 26E

To determine

To sketch: The region bounded by the given curve and visually estimates the centroid location.

To find: The exact coordinates of the centroid (x¯,y¯).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.3, Problem 25E

Chapter 8.3, Problem 27E

Students have asked these similar questions

(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z ≤ 3}. Calculate the integral f(x, y, z) dv. E

(12 points) Let E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}. (a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such that (x, y, z) (psin cos 0, psin sin 0, p cos) € E. (b) (8 points) Calculate the integral E xyz dV using spherical coordinates.

(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}. Calculate the integral y, f(x, y, z) dV.

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 region bounded by the given curve and visually estimates the centroid location. To find: The exact coordinates of the centroid ( x ¯ , y ¯ ) .

Solution Summary: The author sketched the region bounded by the given curve and visually estimated the centroid location.

Concept explainers

Videos

To sketch: The region bounded by the given curve and visually estimates the centroid location.

To find: The exact coordinates of the centroid (x¯,y¯).

Want to see the full answer?

Chapter 8 Solutions

The region bounded by the given curve and visually estimates the centroid location. To find: The exact coordinates of the centroid ( x ¯ , y ¯ ) .

Solution Summary: The author sketched the region bounded by the given curve and visually estimated the centroid location.

Concept explainers

Videos

To sketch: The region bounded by the given curve and visually estimates the centroid location. To find: The exact coordinates of the centroid (x¯,y¯).

Want to see the full answer?

Chapter 8 Solutions

To sketch: The region bounded by the given curve and visually estimates the centroid location.

To find: The exact coordinates of the centroid (x¯,y¯).