Indeterminate Form - Zero Times Infinity

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate the limit for

Solution:

Consider the given equation

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

Since the answer is ∞∙0 which is also another type of Indeterminate Form, it is not accepted in Mathematics as a final answer. We know that any number multiply by zero is always equal to zero but there's an exception, which is infinity. Therefore, we cannot say that infinity times zero is zero. In this type of Indeterminate Form, you cannot use the L'Hopital's Rule because the L'Hopital's Rule is applicable for the Indeterminate Forms like 0/0 and ∞/∞. In this case, let's rewrite the given equation as follows

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

Since the Indeterminate Form at this time is ∞/∞, then we can apply the L'Hopital's Rule to the above equation as follows

Since the trigonometric functions at the above equation are all cancelled and the constant is left, then we cannot substitute the value of x. Therefore,

Triangular Prism Problem

Category: Solid Geometry, Trigonometry

"Published in Newark, California, USA"

The Pennsylvania Railroad found it necessary, owing to land slides upon the roadbed, to reduce the angle of inclination of one bank of a certain railway cut near Pittsburgh, PA, from an original angle of 45º to a new angle of 30º. The bank as it originally stood was 200 ft. long and had a slant length of 60 ft. Find the amount of the earth removed, if the top level of the bank remained unchanged. 

Photo by Math Principles in Everyday Life

Solution:

The given figure is the cross section of a bank. The darker area of the cross section will be removed so that the inclination will be 30º instead of 45º. The darker area is also an obtuse triangle and we need to find the area of that one. Let's start to get the sides of the triangle by analyzing the figure as follows

Photo by Math Principles in Everyday Life

Consider the 45º triangle in order to solve for h and c as follows

Consider the 30º triangle in order to solve for d as follows

The base of a triangle can be calculated as follows

The area of a triangle is also the area of the base is calculated as follows

Finally, the volume of a prism is also the amount of the soil or earth removed from the bank is calculated as follows

Integration - Miscellaneous Substitution

Category: Integral Calculus, Trigonometry

"Published in Newark, California, USA"

Find the integral for

Solution:

You notice that the denominator contains trigonometric functions and we cannot integrate it by simple integration. This is a difficult one because the numerator has no trigonometric functions. If you will use the integration by parts, then the above equation will be more complicated and there will be an endless repetition of the procedure. 

For this type of a function, like the given equation above, we can integrate it by Miscellaneous Substitution. Let's proceed with the integration technique as follows

Let 

From the double angle formula,

Since the given problem has Sine and Cosine functions, then we can get the values of Sine and Cosine functions from Tangent function as follows

Using the figure above that

From the given problem

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

but

Therefore,

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