Explanation Given: The paraboloid z = 3 x 2 + 3 y 2 and the cylinder x 2 + y 2 = 4 . Concept Used: The surface area of the surface given by z = f ( x , y ) over a region Dis given by: S = ∬ D ( f x ) 2 + ( f y ) 2 + 1 d A Calculation: To find the surface area of the given paraboloid z = 3 x 2 + 3 y 2 that lies inside the cylinder x 2 + y 2 = 4 , that is inside the disk of radius 2 centered at the origin, first find the partial derivatives of the given surface. To find the partial derivative of the function, first find the derivative of above function with respect to x , treating y as a constant, it gives f x = ∂ ∂ x ( 3 x 2 + 3 y 2 ) = 6 x Again, differentiating the given plane with respect to y , treating x as a constant, it gives f y = ∂ ∂ y ( 3 x 2 + 3 y 2 ) = 6 y Since the region is the disk of radius 2 centered at origin, so use the polar coordinates to evaluate the integral. The limit of integration in polar coordinates given by 0 ≤ r ≤ 2 , 0 ≤ θ ≤ 2 π So, the surface area of the paraboloid inside the given cylinder is S = ∬ D ( f x ) 2 + ( f y ) 2 + 1 d A = ∬ D ( 6 x ) 2 + ( 6 y ) 2 + 1 d A = ∬ D 36 x 2 + 36 y 2 + 1 d A = ∬ D 36 ( x 2 + y 2 ) + 1 d A = ∫ 0 2 π ∫ 0 2 36 ( r 2 ) + 1 r </

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

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

Publisher: Kendall Hunt Publishing

See similar textbooks

Videos

Question

Chapter 12.4, Problem 10PS

To determine

The surface area of the portion of the paraboloid z=3x2+3y2 that lies inside the cylinder x2+y2=4 .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.4, Problem 9PS

Chapter 12.4, Problem 11PS

Students have asked these similar questions

3. Find the dimension of the largest rectangle (in terms of area) that can be inscribed in a circle of radius V2 cm. 4. Find the point on the line y = 2x – 1 that is closest to the origin. %3D -

CAN PROJECT Metal comes in large, flat sheets. If we ignore waste metal in the manufacturing process, then then to minimize metal we minimize surface area, like on Wednesday. In that case, we got that h = 2r is best. But cans aren't like that! Problem 1. The material for cans is cut from sheets of metal. The cylinder's sides are formed by bending rectangles; these rectangles are cut from the sheet with little or no waste. But if the top and bottom are cut from squares with side length 2r, like in the picture, there's waste metal. When the circles are cut like this, minimize the total metal used to make a can containing 1L.

3. Find the right triangle of largest area inscribed under the parabola y = 4 – r² with its base along the x axis (as shown). o sa

Chapter 12 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

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 surface area of the portion of the paraboloid z = 3 x 2 + 3 y 2 that lies inside the cylinder x 2 + y 2 = 4 .

Solution Summary: The author calculates the surface area of the paraboloid z = 3x2+3y2, which lies inside the cylinder.

Videos

The surface area of the portion of the paraboloid z=3x2+3y2 that lies inside the cylinder x2+y2=4 .

Want to see the full answer?

Chapter 12 Solutions