Introduction to Triple Integral Calculator:
The triple integral calculator with steps is an amazing source for finding the triple integral problems according to the variables in a few seconds. Our tool evaluates the mass and volume of a given area under a closed region in R3 space.
Triple integral has a complicated method, when you evaluate integral's types by hand you face difficulty in getting its solution. In this situation, you need a tool like our triple integration calculator that gives you quick and easy solutions of your given problems.
What is a Triple Integral?
Triple integral is a process of finding the mass and volume of a given function over a bounded region in three-dimensional space. It can calculate the double integral's mass but also the volume of the given region because it involves three variables (x, y, z).
The triple integral is expressed as “∫∫∫” in mathematical calculus. It is known as the multivariable integral method or triple integral. It evaluates the integral three times as per the order of the given function.
Formula of Triple Integral:
The formula of triple integral consists of three functions f (x, y, z) dv and its limits are x1 to x2, y1 to y2, and z1 to z2 in a bounded region.
$$ \iiint_S f(x, y, z)\; dV \;=\; \int_{z_1}^{z_2} \int_{y_1}^{y_2} \int_{x_1}^{x_2} f(x, y, z) dx\; dy\; dz $$
Evaluation Process of Triple Integral Solver:
The volume integral calculator gives the easiest process of integration after the calculation process that gives you a clear understanding about the triple integral solutions. The calculation process of triple integral is given as:
Step 1: The triple integrals calculator first determines the triple integral function to analyze which function is inner, middle or outer function from the given problem.
Step 2: Evaluate the integral with respect to the variable of the inner function, and find limits if the given triple integral function has limits.
Step 3: After solving the inner function, take the integral with respect to the variable of the middle function, and apply limits if the given triple integral function has limits.
Step 4: Lastly, take the integral with respect to the variable of the outer function, and apply limits if the given triple integral function has limits.
Step 5: After evaluating all the triple integral function you get the solution.
Solved Example of Triple integral:
The example of triple integral problem with a solution is given below to let you understand how the 3d integral calculator works.
Example: Evaluate the following,
$$ \int_0^3 \int_0^2 \int_0^1 (xy + z)dx\; dy\; dz $$
Solution:
Identify the given integral and find out the inner function, middle function, and the outer function,
$$ \int_0^3 \int_0^2 \int_0^1 (xy + z)\; dx\; dy\; dz $$
As you can see x is the inner function, y is the middle function and z is the outer function.
Integrate the above function with respect to x and then apply limits.
$$ \int_0^3 \int_0^2 \int_0^1 (xy + z)\; dx\; dy\; dz \;=\; \int_0^3 \int_0^2 \biggr( \frac{1}{2}x^2 y + xz \biggr|_{x=0}^{x=1} \biggr)\; dy\; dz $$
$$ =\; \int_0^3 \int_0^2 \biggr( \frac{1}{2}y + z \biggr)\; dy\; dz $$
Now integrate the middle function with respect to y, and apply limits to get the solution.
$$ \int_0^3 \biggr(\frac{1}{4}y^2 + yz \biggr|_{y=0}^{y=2} \biggr)\; dz $$
$$ \int_0^3 (1+2z) dz $$
Lastly, integrate the outer function with respect to z and apply its limits to get the solution.
$$ =\; \int_0^3 (1 + 2z) dz $$
$$ =\; z + z^2 \biggr|_0^3 \;=\; 12 $$
Therefore the given triple integral problem solution is 12.
How to Use the Triple Integration Calculator?
The triple integral calculator has a simple design that helps you to calculate the triple integral questions easily. Before adding the integral function as an input for triple integral solutions, you must follow some of our guidelines. These guidelines are:
- Enter your triple integral problem in the input field.
- Select the integration variables for triple integral problems.
- If the given integral problem is definite then add the limit values in the respective box otherwise keep it empty.
- Recheck your triple integral problem before hitting the calculate button to start the calculation process in the triple integral solver.
- Click on the “Calculate” button to get the desired result of your given triple integral question.
- If you want to try our volume integral calculator for the first time then you can use the load example.
- Click on the “Recalculate” button to get a new page for solving more triple integral problems to get solutions.
Final Result of Triple Integrals Calculator:
The 3d integral calculator gives you the solution to a given triple integral problem when you add the input into it. It provides you with solutions to the triple integral problem. It may contain the following:
- Result Option
When you click on the result option then it provides you with a solution for the triple integral questions.
- Possible Step:
When you click on the possible steps option it provides you with the step by step solution of triple integral problems in detail.
Useful features of Triple Integral Calculator With Steps:
Triple integration calculator has many useful features that you obtain whenever you use it to calculate triple integral problems. These features are:
- It is an easy to use tool that can operate through electronic devices like laptops, computers, mobile, tablets, etc with the help of the internet.
- Triple integral solver is a free tool so you can use it to find the triple integral problems.
- Our tool saves the time and effort that you consume in doing lengthy calculations of triple integral.
- You can use the volume integral calculator to practice so that you get in-depth knowledge about this method.
- It is a trustworthy tool that provides you with accurate solutions of triple integral problems.
- Triple integrals calculator provides you solutions of triple integral questions with a complete step by step process.
