Inspecting Integrating Factors

Category: Differential Equations, Integral Calculus, Algebra

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above

Expand the above equation and regroup as follows

Divide both sides of the equation by y², we have

The grouped terms in parenthesis can be simplified as an exact differential as follows

Integrate on both sides of the equation, we have

Multiply both sides of the equation by y, we have

Therefore, the final answer is

Right Circular Cylinder - Sphere

Category: Solid Geometry

"Published in Newark, California, USA"

Two balls, one 6 in. in diameter and the other 4 in. in diameter are placed in a cylindrical jar 9 in. in diameter, as shown. Find the volume of water necessary to cover them.

Photo by Math Principles in Everyday Life

Solution:

The first that we have to do is to analyze and label the given figure as follows

Photo by Math Principles in Everyday Life

Consider a right triangle between the centers of two spheres and use Pythagorean Theorem in order to solve for x as follows

The height of a right circular cylinder is

The volume of a right circular cylinder is

The volume of a big sphere is

The volume of a small sphere is

Therefore, the volume of water necessary to cover the two balls is

or you can give the value of pi as follows

Derivative - Trigonometric Functions

Category: Differential Calculus, Trigonometry

"Published in Suisun City, California, USA"

If n is a positive integer, prove that

Solution:

Consider the given equation above

We will use the left side of the equation to prove the right side of the equation as follows

Apply the derivative by product and then trigonometric functions, we have

Separate their common factor, we have

Therefore,