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

**Solution.**

**Method (i).**

.

Therefore

.

Hence .

**Method (ii).**

Since , we get

Assume y=sx, we get

Therefore,

**Question 2.** Calculate the value of

**Solution.**

**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

**Solution. **

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

**Solution.**

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

**Solution.**

**Method (i). **Use the polar coordinate.

**Method (ii). **Make the substitution , then

The region is and

That is equivalent to and

The integration is

### Like this:

Like Loading...

*Related*