Finding Equation - Parabola

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a parabola with horizontal axis, vertex on y axis, and passing through the points (2, 4) and (8, -2).

Solution:

To illustrate the problem, let's plot all the given items and sketch the parabola in the rectangular coordinate system as follows

Photo by Math Principles in Everyday Life

As you can see in the figure, we can have two equations of parabola based on the given items in the problem. Since the axis of the parabola is horizontal and it opens to the right, the equation of a parabola in standard form is

The vertex of a parabola is located in y-axis, the coordinates of the vertex is (0, k). The above equation becomes

If (2, 4) is one of the points of a parabola, substitute the values of x and y to the above equation

If (8, -2) is one of the points of a parabola, substitute the values of x and y to the above equation

Equate

Multiply both sides of the equation by 32 and solve for the value of k

Divide both sides of the equation by 3

Equate each factor to zero and solve for the value of k

If 

then

Solve for the value of a, we have

Therefore, the equation of a parabola in standard form is

If

then

Solve for the value of a, we have

Therefore, the equation of a parabola in standard form is

Integration Procedure - Algebraic Susbstitution

Category: Integral Calculus, Algebra, Differential Calculus

"Published in Newark, California, USA"

Evaluate

Solution:

If you examine the given equation, the denominator is complicated because it has a variable and a radical function as well. The differential at the numerator is not match with the differential of the denominator. Therefore, we cannot integrate the given equation by simple integration. If you will integrate the given equation by parts, then the equation will be more complicated and there's no end in integration as well. We have a procedure for that type of integration which is called an algebraic substitution. Let's consider the given equation

Let

Substitute these values to the given equation, we have

But

The equation becomes

Since the first term is indeterminate form, then we have to evaluate the limit of the first term since ∞/∞ is not accepted as a final answer in Mathematics. Let's evaluate the limit of the first term as follows

Apply the L'Hopital's Rule

Therefore,

Rationalize the denominator by multiplying both sides of the fraction by the conjugate of the denominator, we have