Question

hi can anyone help with with this problem:
the square region with the vertices (0, 2), (2,0), (4,2) and (2,4) is revolved about the x-axis to generate a solid. Using the theorem of pappus, find both the volume and the surface area of the solid.

Answer 1

I am assuming that you are referring to Pappus's Centroid Theorem. If so then:

Let's call the side length of the square S.
S = sqrt(2^2 + 2^2)
S = Sqrt(8) = side length of square

Since this square is rotated form a solid, the solid formed is a cylinder with a radius of sqrt(8). Since the surface of reflection is formed by the rotation of a square, the cylinder must then also have the height of sqrt(8).

Now by Pappus' Centroid Theorem,

SA = 2pi*s*x Here in the cylinder's case, s = h (h represents height) x = r (r represents radius)

Therefore: SA = 2pi*h*r
= 2pi*sqrt(8)*sqrt(8)
= 16pi

Now, according to Pappus' Centroid Theorem, the volume of the cylinder is:

V = 2pi*A*X Here in the cylinder's case, A = hr (Where h is the height and r is the radius)
X = 1/2r (Where r is the radius)

Therefore: V = 2pi*hr*1/2r
= pi*r^2*h
= pi * (sqrt(8))^2 * sqrt(8)
= 8pi*sqrt(8)

Answer 2

would the radius be half the hypotenuse instead cause when i inputted the answer and it was wrong, and does it really create a cylinder since the sides are not curved?

Answer 3

pappus theorems:
\[\Large {(1) V=2\pi \overline{x} A\\(2) S_A=2\pi\overline{x}P}\]
where \(\overline{x}=\) distance of the centroid to the axis of revolution
\(A=\) area of the region
\(P=\) perimeter of the region