Triple integrals. Dear friends, today I’ll show how to evaluate triple integrals. Have a look!!

**How to evaluate triple integrals**

**Some solved examples of triple integrals**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

**Example 1**

According to Stroud and Booth (2013)*, “Evaluate

”

**Solution**

Now here I have to evaluate the triple integral

So let’s say

First of all, I’ll integrate for . Then I’ll integrate for . Finally, I’ll integrate for . And, I’ll use the * formulas in integration* for that.

**Step 1**

Therefore it will be

Next, I’ll substitute the limits. Now here the lower limit is . And the upper limit is .

Thus will be

Then I’ll simplify it to get

Now I’ll integrate it for .

**Step 2**

So it will be

Then I’ll substitute the limits. Now here the lower limit is . And the upper limit is .

Thus will be

Then I’ll simplify it to get

Next, I’ll integrate it for .

**Step 3**

So it will be

Then I’ll substitute the limits. Now here the lower limit is . And the upper limit is .

Thus will be

Then I’ll simplify it to get

Now I’ll take out the common terms. And that gives the value of as

Hence I can conclude that this is the answer to the given example.

Now I’ll give you another example.

**Example 2**

According to Stroud and Booth (2013)*, “Evaluate

”

**Solution**

Now here I have to evaluate the triple integral

So let’s say

First of all, I’ll integrate it for . Then I’ll integrate for . Finally, I’ll integrate it for .

And just like example 1, here also I’ll use the standard formulas in integration.

**Step 1**

Therefore it will be

Next, I’ll substitute the limits. Now here the lower limit is . And the upper limit is .

Thus will be

Then I’ll simplify it to get

Now I’ll integrate it for .

**Step 2**

So it will be

Then I’ll substitute the limits. Now here the lower limit is . And the upper limit is .

Thus will be

Then I’ll simplify it to get

Next, I’ll integrate it for .

**Step 3**

So it will be

Then I’ll substitute the limits. Now here the lower limit is . And the upper limit is .

Thus will be

Then I’ll simplify it to get

Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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