Pappus Theorem surface area

In summary, the volume and surface area obtained by rotating the given area can be calculated by breaking down the shape into its individual components and using the appropriate formulas to find the total volume and surface area.
  • #1
noname1
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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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  • #2
.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.
 

1. What is Pappus Theorem and what does it have to do with surface area?

Pappus Theorem is a mathematical principle that relates the volume of a solid of revolution to the surface area of its generating curve. In other words, it allows us to calculate the surface area of a 3D object by using its 2D cross-section and the distance traveled by the cross-section as it rotates around an axis.

2. How do you use Pappus Theorem to find the surface area of a solid?

To use Pappus Theorem, you need to know the distance traveled by the generating curve as it rotates around an axis (known as the centroidal distance) and the area of the 2D cross-section. The surface area can then be calculated by multiplying the centroidal distance by the area of the cross-section.

3. Can Pappus Theorem be applied to any type of solid object?

Yes, Pappus Theorem can be applied to any solid object that can be generated by rotating a 2D shape around an axis. This includes objects like spheres, cones, cylinders, and more complex shapes.

4. How accurate is Pappus Theorem in calculating surface area?

Pappus Theorem provides an accurate approximation of the surface area of a solid object. However, like any mathematical theorem, it is dependent on the accuracy of the measurements and assumptions used in the calculation.

5. Can Pappus Theorem be used to find the volume of a solid?

No, Pappus Theorem only applies to the surface area of a solid. To find the volume of a solid, you would need to use other mathematical principles such as the disk method or shell method.

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