whatlifeforme said:Homework Statement
Use Pappus' Theorem for surface area and the fact that the surface area of a sphere of radius c is 4(pi)c^2 to find the centroid of the semicircle x= sqrt ( c^2 - y^2)Homework Equations
S = 2 (pi) * p * L
where s=surface area; p=distance from axis of revolution; L= length of the arcThe Attempt at a Solution
1. centroid of semicircle. should i put the equation in circle form, and attempt to solve from that?
whatlifeforme said:i'm still lost.
whatlifeforme said:S = 2 (pi) * p * L
where s=surface area; p=distance from axis of revolution; L= length of the arc
Pappus' Theorem (surface area) is a mathematical theorem named after the Greek mathematician Pappus of Alexandria. It states that the surface area of a three-dimensional object can be found by multiplying the length of its circumference by the distance traveled by its centroid while rotating around an axis.
Pappus' Theorem (surface area) is commonly used in calculus and geometry to find the surface area of objects with curved surfaces, such as spheres, cones, and tori. It can also be used to calculate the surface area of more complex shapes by breaking them down into simpler components.
In order to use Pappus' Theorem (surface area), the object must have a constant cross-sectional area and its centroid must travel a constant distance while rotating around an axis. Additionally, the axis of rotation must be parallel to the cross-sectional area.
Yes, Pappus' Theorem (surface area) is limited to finding the surface area of objects with continuous and symmetrical cross-sections. It also cannot be used to find the surface area of objects with changing cross-sectional areas or non-uniform rotations.
No, Pappus' Theorem (surface area) can only be applied to objects with rotational symmetry. This means that the shape of the object must remain the same after being rotated around an axis. Examples of shapes that can be used with Pappus' Theorem include cylinders, spheres, and cones.