In the tutorial 7, we will learn to calculate the integration of two variables, reverse the order of integration and polar coordinate.
The formulas of polar coordinate are , , where and .
Question 1. The application of polar coordinate. Calculate the value of
Since , we get
Assume y=sx, we get
Question 2. Calculate the value of
Method (i). Leibniz Integration Rule.
Here denotes the partial derivative of with respect to the variable .
In the question, assume .
Making use of L’Hospital Rule, we have
Method (ii). Reverse the order of integration.
The integration domain is and . It is same as and .
Question 3. MA1505 2010-2011 Semester 2, Question 6(b).
Let R be a region of xy-plane, find the largest possible value of the integration
Since we want to find the largest possible value, then we must guarantee that on the region R, the function is non-negative. That means the region R is . i.e. . Therefore, we should calculate the integration
Question 4. is a real interval, calculate the maximum value of
To calculate the maximum value of the integration, the maximal interval Therefore, the maximum value of the integration is
Qustion 5. Calculate the multiple integration
Method (i). Use the polar coordinate.
Method (ii). Make the substitution , then
The region is and
That is equivalent to and
The integration is