Surface of a Sphere

Let's derive the formula for the surface of a sphere. A sphere is a revolution of a circle. Let there be a circle whose function is x²+y²=r². Then its center is (0,0) and r is its radius.

When we revolve the circle about the x-axis, we get a sphere. Since we already know the formula for the surface of a revolution, we can find the surface area of a sphere as follows:

Differential of Arc Length

I was trying to derive the formula for obtaining the surface area of a sphere, but I kept making mistakes whereby I chopped the surface in dx or dy. When you calculate the arc length of a curve or the surface area of a volume, you have to chop it in lengths of the slope, not the width or the height.

To compute the length of an arc of a curve, we need to define the differential of arc length. If we zoom in on a small section of a curve, we find that the length of a curve s is the length of the slope of a right triangle whose base is dx and whose height is dy.

Therefore,

Surfaces of Revolution

Using the differential of arc length, we can find the area of the surface generated by rotating a curve C about the x-axis or y-axis. When a curve of function y=f(x) is rotated about the x-axis, the surface area of the generated solid can be found with the following formula:

When a curve of function y=f(x) is rotated about the y-axis, the surface area of the generated solid can be found with the following formula:

Calculus Formulas: Finding Anti-Derivatives

Properties of Definite Integrals

Basic Properties of Integrals

Derivatives of Definitive Integrals

Table of Integrals

Calculus Formulas: Finding Derivatives

Basic Formulas of Derivatives

For any two differentiable functions f(x) and g(x):

1. f(x)=c (c is a constant), then f'(x)=0
   
2. g(x)=cf(x) (c is a constant), then g'(x)=c·f'(x)
   
3. f(x)=xⁿ (n is a rational number), then f'(x)=nx^n-1
   
4. y=f(x)±g(x), then y'=f'(x)±g'(x)
   
5. y=f(x)·g(x) then, y'=f'(x)·g(x)+f(x)·g'(x)
   
6. y=f(x)/g(x) (g(x)≠0) then, y'=(f'(x)g(x)-f(x)g'(x))/g²(x)
   

Derivatives of Various Functions

1. y=e^x then, y'=e^x
   
2. y=a^x then, y'=a^x·ln(a)
   
3. y=ln(x) then, y'=1/x
   
4. y=log_ax then, y'=1/(x·ln(a))
   
5. y=sin(x) then, y'=cos(x)
   
6. y=cos(x) then, y'=-sin(x)
   
7. y=tan(x) then, y'=1/cos²(x)
   
8. y=arctan(x) then, y'=1/(x²+1)
   
9. y=arcsin(x) then, y'=(1-x²)^-0.5
   

Derivative of Composite Functions

1. y=f(u), u=g(x) then, dy/dx=dy/du · du/dx
   
2. y=f(ax+b) then, y'=af'(ax+b)
   
3. y=fⁿ(x) then, y'=n·f^n-1(x)·f'(x)