Circle - Circumference Derivation

Category: Integral Calculus, Differential Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

If the equation of a circle is x² + y² = r², prove that the circumference of a circle is C = 2πr.

Solution:

To illustrate the problem, let's draw the graph of a circle as follows

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If the equation of a circle is

We can rewrite the above equation as a function of x as follows

Take the derivative of the above equation with respect to x as follows

To get the length of a curve or circumference of a circle, consider only a quadrant as follows

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The length of a curve is given by the formula

Substitute the value of ^dy/_dx to the above equation, we have

Therefore, the circumference of a circle is

Maximum Minimum Problem, 3

Category: Differential Calculus, Trigonometry, Algebra

"Published in Suisun City, California, USA"

Find the maximum and minimum values of the function

Solution:

Consider the above equation

Take the derivative of the above equation with respect to x as follows

Since we want to solve for the maximum and minimum values, equate the above equation to zero as follows

To solve for the value of Sin x, use the Quadratic Formula as follows

If you choose the positive sign

Therefore,

If you choose the negative sign

Therefore,

More Integration Procedures, 3

Category: Integral Calculus, Algebra

"Published in Suisun City, California, USA"

Evaluate the integral for

Solution:

The above equation has two radicals with different indexes. We need to substitute each radicals with another variable with their common index as follows

Let

Substitute the above values to the given equation, we have

Since the exponent at the numerator is greater than the exponent at the denominator, then we need to do the division of polynomial as follows

The above equation becomes

but

Therefore, the final answer is