Pappus Theorem surface area

  1. 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
     

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