Multiple Integrals Tutorial and Double Integral Solutions with Examples

Multiple Integrals

Multiple Integrals means during integration of one variable can be extended to the functions of more than one variable. The generalizations of define integrals is known as Multiple integral. Multiple integral covers several of topics such as:

1. Double Integrals
2. Iterated Integrals
3. Double Integrals in polar coordinates
4. Triple Integrals
5. Triple integrals in cylindrical coordinates

Note: Triple Integrals in spherical coordinates and Double Integrals over general regions.

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Best Quality Double Integral Problems and Detailed Solutions

Notation of Double Integral





This shows the double integral, over the reason of z of f(x,y).

Working Method of double integral








First Integral (A), Integrate with respect to y and x is assumed as a constant and evaluate the result between the limit y = f1(x) and y f2(x). After this we integrate the result of (A) with respect to x between the limit x=a and x=b.

Examples of Double Integrals
(1)








Step - 1
First we Integrate with respect to y and x is assumed as a constant between the limit y=0 and y=x.



























Example-2

y = 3x

y = 5-2x

y = 0


























Solving these three equations

y = 3x

y = 5 – 2x

y = 0

we get lines cut at (0,0) , (5/2, 0) and (1, 3)

thus limit of x will vary from x = 0 to 5/2.

And limit of y will vary from y = 3x to y = 5 – 2x














Area A = 25/8 Ans.

In above examples we have provided the procedure of double integral problems with examples and if you are still confuse to solve double integral problems then do let us know. We have very good team of tutors with 5 years of experience. We have done thousands of assignments related to double integral problems. I would be very happy to provide you assignment help. Please do send us your queries or assignment. We will provide you details solution with accuracy and our cost would be very reasonable.

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How to Calculate Net Present Value with Example?

Net present value is the difference between present value (PV) of cash inflows means initial investment and the present value (PV) of cash outflows mean future value. Net Present value in terms of money can be positive, negative and zero.

Following are the examples of NPV:

Example-1
John have $200,000 and he want to invest money in land. John thinks that the future value of the land would be $350,000. Then

Net Present Value (NPV) = future value of land – present value of land

• NPV= $350,000 - $200,000
• NPV = $150,000 ( positive)

From the above example we can conclude that John will get very good rate of return.

Example 2
Maria have $200,000 and he want to invest money in land. maria thinks that the future value of the land would be $190,000. Then

Net Present Value (NPV) = future value of land – present value of land

• NPV= $190,000 - $200,000
• NPV = -$10,000 ( Negtive)

From the above example we can conclude that the future value of the cash flows is less than the initial investment and Maria will get very less rate of return

Example: 3
Paul have $200,000 and he want to invest money in land. Paul thinks that the future value of the land would be $200,000. Then

Net Present Value (NPV) = future value of land – present value of land

• NPV= $200,000 - $200,000
• NPV = 0 ( Zero)

From the above example we can conclude that the future value of the cash flows is equally to initial investment and Paul will get the initial investment money.

General formula of NPV



Where T = Time period Ct = Cash inflow Co = Cash Out flow

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Purpose of Accounting and What is a Double Entry System?

What is Accounting
In business, we sell and buy lots of goods and these sell and buy cost, we writes in accounting books. To written these entries in accounting book are called accounting.

Importance of Accounting

• Work as a profit
• As a Result
• For decision making

What is a Double Entry System?
In every business when we buy or sell goods, we see in each transaction there will be two entries: one is Debit entry and other one is Credit entry. When anything coming in business we used Debit side entry and anything going from business we used credit side entry. So both transition mean debit and credit are recorded together with equal amount is Know as Double entry system.

Example:
John purchased goods in cash and cost of goods is $750000.
We see that goods are coming in business and cash is going. So goods should be written in debit side and cash should be write in credit side and this transition in Double entry system will be written as follows:



Advantages of Double Entry System

1. Correct calculation so there will no error
2. Easy to check profit and loss in business then owner take decision according to that.
3. Transitions of any entry, we can find in few minutes
4. Money Owed by debtors and creditors, we can check in minutes.
5. Time saving

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