Explanation Given: f ( x , y ) = x y x 2 + y 2 + 1 ; R : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 Calculation: Consider a line f ( x , y ) f ( x , y ) = x y x 2 + y 2 + 1 And rectangle R such that R : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 Find the volume of solid bounded below by rectangle R in the x y plane and above by the graph of z = f ( x , y ) So volume will be evaluated by finding the following integral ∫ ∫ R f ( x , y ) d A This is same as; ∫ ∫ R f ( x , y ) d A = ∫ 0 1 ∫ 0 1 ( x y x 2 + y 2 + 1 ) d y d x So evaluate ∫ 0 1 ∫ 0 1 ( x y x 2 + y 2 + 1 ) d y d x First keeping x as constant integrate with respect to y ∫ ∫ R f ( x , y ) d A = ∫ 0 1 ∫ 0 1 ( x y x 2 + y 2 + 1 ) d y d x = ∫ 0 1 ( ∫ 0 1 x y x 2 + y 2 + 1 d y ) d x Since ∫ c b ∫ c d f ( x , y ) d y d x = ∫ c b ( ∫ c d f ( x , y ) d y d x ) Further solving Let t = y 2 , differentiate t with respect to y d t d y = 2 y 2 y d y = d t y d y = d t 2 And when y = 0 then t = 0 and when y = 1 then t = 1 So the integral ∫ ∫ R f ( x , y ) d A = ∫ 0 1 ∫ 0 1 ( x y x 2 + y 2 + 1 ) d y d x becomes ∫ ∫ R f ( x , y ) d A = 1 2 ∫ 0 1 ( ∫ 0 1 x x 2 + y 2 + 1 d t d x ) = 1 2 ∫ 0 1 ( x x

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

See similar textbooks

Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Videos

Question

Chapter 12.1, Problem 37PS

To determine

To find: the volume of the solid bounded.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.1, Problem 36PS

Chapter 12.1, Problem 38PS

Students have asked these similar questions

6. Determine the absolute maximum and minimum values of f(x, y) = 2x² + 3y² on the rectangular region where -1 ≤ x ≤ 1 and -1 ≤ y ≤ 1.

1. The area bounded by the curves y x2,y 0, x = 2,x 5 is 39. 2. The area bounded by the curves y = x,y 0, x 1,x 3 is 19. 3. The area bounded by the curves y = 4x – x², y = 0,x = 1,x = 3 is. 4. The area bounded by the curves x = 1+ y2,x = 10 is 36. 5. The area bounded by the curves y = 25 - x2, 256x = 3y2, 16y = 9x2 is . %3D 104V13-125 6. The length of the arc of the curve y3 = 8x? from x = 1 to x = 8 is - 27 %3D 7. S, (x2 – x3 + 3)dx = 10

1. A square plate with sides of unit length has its center at the origin of an x-y coordinate plane; the sides are parallel to either the x- or the y- axis. A hole of radius r is removed from the plate the center of this hole is on the x-axis at a distance r from the left side of the plate. The centroid of the remaining piece has an x-coordinate equal to c. The value of r that maximizes c satisfies the equation: Tr³ – 3r + 1 = 0 a. Draw the figure and label the parts. b. Find the real root r of the cubic equation by the SECANT METHOD. Use e = 0.0001 and starting values 0.3500 and 0.4500.

Chapter 12 Solutions

Calculus

Ch. 12.1 - Prob. 1PS Ch. 12.1 - Prob. 2PS Ch. 12.1 - Prob. 3PS Ch. 12.1 - Prob. 4PS Ch. 12.1 - Prob. 5PS Ch. 12.1 - Prob. 6PS Ch. 12.1 - Prob. 7PS Ch. 12.1 - Prob. 8PS Ch. 12.1 - Prob. 9PS Ch. 12.1 - Prob. 10PS

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 volume of the solid bounded.

Solution Summary: The author explains how to find the volume of the solid bounded below by rectangle R in the xy plane and above by the graph.

Concept explainers

Videos

To find: the volume of the solid bounded.

Want to see the full answer?

Chapter 12 Solutions