How Does Pappus' Theorem Calculate Rotational Surface Areas?

[noname1]

determine the volume and surface area obtained by rotating the area

i know how to obtain the volume but i am confused on how to obtain the area, i have the solutions but i want to learn how to do it

i know that the area of the rectangle is 1500 and the semicircle is -353.42 so the total area is 1146.57

i know that the rectangle x bar is 15, y bar is 25, so xbar * area = 22500 and ybar*area=37500


i know that the semicircle x bar is 23.63, y bar is 30, so xbar * area = -8351.3 and ybar*area= -10602.6

so total xbar area is 14148.6 and ybar area 26897.4

now volume on ybar is 2*pi*26897.4=169001.35
and volume on xbar is 2*pi*14148.6=88898.7

but i see to obtain the area on on the x-axis is

A= 2*pi*(112.5+1413.7+237.5+1250+1500)

and y is 2*pi*(450+450+942.47+150+450)

my question is where did these numbers inside the parenthesis came from, i would appreciate if someone could explain to me how these numbers were obtained


thanks in advance

 

[lippyka]

.The numbers in the parentheses represent the lengths of the various parts of the shape created by rotating the given area. For example, on the x-axis, you have a rectangle with a length of 1500 and a semicircle with a radius of 23.63. The length of the rectangle is 1500 and the length of the semicircle is 2*pi*radius = 2*pi*23.63 = 147.67. So the total length on the x-axis is 1500 + 147.67 = 1647.67. This is then broken down into the individual components: 112.5 (length of the left side of the rectangle), 1413.7 (length of the bottom side of the rectangle), 237.5 (length of the arc of the semicircle), 1250 (length of the right side of the rectangle), and 1500 (length of the top side of the rectangle). The same process is used to calculate the lengths on the y-axis.