Integration - Trigonometric Substitution

Category: Integral Calculus, Trigonometry, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

If you examine the given equation, we cannot integrate it by simple integration. Since the given equation has √ 4 - x2 , then we have to use the Trigonometric Substitution. Trigonometric Substitution is applicable if a function contains √ a2 - x2 , √ a2 + x2 , and √ x2 - a2 . 


First, draw a right triangle to represent √ 4 - x2  as follows

Photo by Math Principles in Everyday Life

Let

Substitute these values to the given equation, we have

Substitute the values of θ, Sin θ, and Cos θ to the above equation, we have

Substitute the value of the limits to the above equation, we have

Therefore,

Volume - Cube, Given Diagonal

Category: Solid Geometry, Plane Geometry

"Published in Newark, California, USA"

A diagonal of a cube joints two vertices not in the same face. If the diagonals are 4√3 cm. long, what is the volume?

Solution:

To visualize the problem, let's draw the figure as follows

Photo by Math Principles in Everyday Life

We know that all sides of a cube are equal because all faces of a cube are square. All sides of a cube are perpendicular to each other. A diagonal is a line segment that connects the two opposite vertices of a cube. There are 4 equal diagonals in a cube: AG, CE, BH, and FD.

How do you get the length of a diagonal of a cube if one side of a cube is given? Here's the procedure in getting the length of a diagonal of a cube as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem

After we get the diagonal of a base, we can finally get the diagonal of a cube as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem

The length of a diagonal of a cube is equal to the length of a side of a cube times square root of three. From the given word problem that if the length of a diagonal of a cube is 4√3 cm., then the length of a side of a cube will be 

Finally, we can get the volume of a cube as follows

Ratio, Proportion - Elongation Problem

Category: Algebra, Strength of Materials

"Published in Newark, California, USA"

The elongation of any metal is directly proportional to the product of the applied force F and the length L, and inversely proportional to the beam cross-sectional area. A steel bar 2 square inches in cross section and 20 inches in length is elongated by 0.03 inches when a force of 10,000 lbs. was applied to it. A certain member of the same metal whose length is 3 feet is allowed a maximum elongation of 0.5 inch when subjected to a force of 18,500 lbs. Compute the minimum permissible area of the member.

Solution:

To illustrate the problem, let's draw the figure as follows

Photo by Math Principles in Everyday Life

As you can see in the figure that when you applied a force F at the end of a metal with length L, there will be an elongation with length ∆L. From the first statement of the word problem, "the elongation of any metal is directly proportional to the product of the applied force F and the length L, and inversely proportional to the beam cross-sectional area," the working equation can be written as follows

and

Combining the two equations above, we have

or

where

     ∆L = elongation of a metal
       k = proportionality constant
       F = applied force
       L = length of a metal
       A = cross sectional area of a metal

From the second statement of a word problem, "a steel bar 2 square inches in cross section and 20 inches in length is elongated by 0.03 inches when a force of 10,000 lbs. was applied to it," substitute the given items to the working equation in order to get the value of k as follows

or

From the third statement of a word problem, "a certain member of the same metal whose length is 3 feet is allowed a maximum elongation of 0.5 inch when subjected to a force of 18,500 lbs," substitute the given items to the working equation in order to get the value of A as follows

Therefore, the minimum permissible area of the member is 0.1998 in².