Find the area of the surface in the first octant cut from the cylinder y equals two thirds z Superscript 3 divided by 2 by the planes x equals 3 and y equals 18.

A = 339.531 units^2  

Step-by-step explanation:

Given:-

- The surface of consideration is bounded by the following:

                   Cylinder :  

                   Planes : x = 3 , y = 18

Find:-

Find the area of the surface in the first octant

Solution:-

- It is better to sketch the given curves. Taking (y-z) plane on the sketch paper with x - axis out of the page.

- We see that a surface bounded with 5 faces. We will investigate each face one by one.

Face 1 : This face that lies on the y-z plane defined by the curve of the cylinder and y = 18, and z = x = 0.

Face 2: This face has the same area as Face 1. This lies on the plane x = 3 defined by the curve and y = 18, z = 0.

                                   A1 = A2 = 97.19994 units^2

Face 3 : This face lies on the plane where y = 18 parallel to x-z plane. Representing a rectangle, defined by plane y = 18 where it intersects the cylinder and z = 0 ... Denoting width of rectangle. While the length is defined by the intersection of planes (y = 18 and x = 3).

                                    A3 = Length*Width

                                    A3 = (3)*(9)

                                    A3 = 27 units^2

Face 4 : This face lies on the plane where z = 0 parallel i.e x-y plane. Representing a rectangle, defined by plane y = 18 where it intersects the z = 0 denoting length, while the width is defined by the intersection of planes (x = 3 and z = 0).

       The area of Face 4 can be calculated directly from rectangle formula:

                                  A4 = Length*Width

                                  A4 = (3)*(18)

                                  A4 = 54 units^2

Face 5 : This face is curved in nature defined by the cylinder. Which starts from plane x = 0 and limits at x = 3. We will determine the Area of this face by first computing the arc length (L) of the cylinder curve defined by:

                            A5 = L*( 3 - 0 )

                            A5 = (21.37736)*(3)

                            A5 = 64.13208 units^2

- The total surface area of the enclosed body is:

                           A = A1 + A2 + A3 + A4 + A5

                           A = 2*(97.1994) + 27 + 54 +64.13208

                           A = 339.531 units^2                  

i would have to say it is c.

560

step-by-step explanation:

giggidy