Integration - Algebraic Substitution, 2

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above

This type of integration cannot be integrated by simple integration. We have to use the technique of integration procedures. If we will use the Integration by Parts, the above equation will be more complicated because it contains radical equation. In this type of integration, we have to use the Algebraic Substitution as follows

Let

So that 

Substitute the above values to the given equation, we have

but 

Therefore, the above equation becomes

Rationalize the denominator so that the radical equation at the denominator will be eliminated as follows

Divide the numerator and denominator by x², we have

Substitute the values of limits and therefore, the final answer is

Note: If you are getting the area bounded by the curves, then the sign must be expressed in the absolute value. The area is always positive. If you interchange the position of the limits in the proper integral, then the sign of the final answer will be change.

Partial Differentiation - Total Derivatives

Category: Differential Calculus

"Published in Newark, California, USA"

Find ^du/_dt for the following functions:

Solution:

Consider the given equation above

Since the first function has three variables, then we have to take the partial derivatives of a function with respect to x and y as follows

Next, take the derivative of the other functions with respect to x since they have only one variables as follows

Therefore,

Substitute the values of x and y in the above equation to obtain the final equation in terms of u and t as follows