Introduction to Double Integral Calculator:
Double integral calculator with steps is an online tool that helps you to find the double integral of two variables in two dimensions. It is used to evaluate the average value of area and volume under a surface within two variables.
It is a learning tool that helps you to teach your children, and students about the double integral method in just one way by sitting at home without going anywhere.
What is Double Integral?
Double integral is a process of finding an area over a defined region in two-dimensional space in calculus. It is called the double integral or double iteration method. It is denoted with the” ∫∫” symbol in math.
It has two variables, which means you multiply the given function with respect to one variable by keeping the second function constant. When you calculate the integral for the second function the first variable behaves as constant. Double integral functions can be definite or indefinite integral, it depends on the nature of the given function for integration.
Formula of Double Integral:
The double integral formula consists of two functions in x and y and you can integrate the function two times. First, integrate the function with respect to x (inner function) and then with respect to y because it is the outer function.
$$ \int_R f(x,y)dA \;=\; \int_a^b \int_c^d f(x,y)dy\; dx $$
Evaluation Process of Iterated Integral Calculator:
Double integration calculator can evaluate the double integral problem in the simplest way that can be understandable to everyone. Because it has an advanced feature that allows you to add any type of problem, it will provide you with a solution.
Step 1: Identify the given double integral function to see which is the inner function and outer function of integration.
Step 2: Write the given integral function in its inner or outer order of function for integration.
Step 3: Now integrate the inner function with respect to its variable. If it has limits then apply limits and get solutions. During the integration of the inner function, the outer function behaves as a constant.
Step 4: After integrating the inner function, integrate the outer function with respect to its variable. If it has a limit then apply limits to get its solution.
Step 5:
Lastly, after solving both the inner and the outer function you get the solution of the double integral problem.
Solved Example of Double Integral:
Let us see an example of double integral with a solution to see how our multiple integral calculator evaluates integral in steps.
Example: Find the volume V under the given plane,
$$ z \;=\; 8x + 6y \; over\; the\; rectangle\; R \;=\; [0,1] \times [0,2] $$
Solution:
In this function, x is the inner function and y is the outer function. According to the given condition double integral can be written as:
$$ V \;=\; \int_0^2 \int_0^1 (8x + 6y)dx\; dy $$
As x is our inner function then integrate the above function with respect to x and then apply the limit to the solution we get as,
$$ =\; \int_0^2 (4x^2 + 6xy \biggr|_{x=0}^{x=1} ) dy $$
$$ =\; \int_0^2 (4 + 6y) dy $$
Then again take the integral with respect to y, apply the limit, and get the volume of the given function.
$$ =\; \int_0^2 (4 + 6y) dy $$
$$ =\; 4y + 3y^2 \biggr|_0^2 $$
$$ 20 $$
How to Use Double Integral Solver?
Multivariable integral calculator has a simple design that makes it easy for you to understand how to use it for the evaluation of a double integral. You just need to follow our instructions that are given as:
- Enter the double integral function you want to calculate in the input field.
- Choose the integral variable.
- If the given function is in definite integral form then add the limit of both the variables.
- Review the given integral function of double integrals calculator before hitting the calculate button to start the evaluation process.
- The “Calculate” button to get the result of your given double integral problem.
- If you are trying our multiple integral calculator for the first time then you can use the load example to learn more about this method.
- The “Recalculate” button to get a new page for finding more example solutions of double integral problems.
Final Result of Double Integral Calculator With Steps:
The iterated integral calculator gives you the solution from a given double integral problem when you add the input to it. It included as:
- Result Option:
When you click on the result option the double integration calculator gives you a solution of the double integral problem.
- Possible Steps:
When you click on it, this option will provide you the step by step solution of the double integral calculations.
Advantages of Using Multiple Integral Calculator:
Double integral solver provides you with many advantages that help you to calculate double integral problems that give you solutions without any difficulty. These advantages are:
- Multivariable integral calculator is a free tool so you can use it for free to find the double integral without spending.
- It is a manageable tool that can manage various types of double integral problems.
- When you use it for practicing, our double integrals calculator helps you to get conceptual clarity of double integral.
- It saves the time that you consume on the calculation of the double integral problems.
- Double integral calculator with steps is a reliable tool that provides you with accurate solutions whenever you use it to calculate the double integral problem.
- Iterated integral calculator provides the solution without imposing any condition of signup. It means you can use it multiple times.
