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Six Ways To Write The Same Iterated Triple Integral

Six Ways To Write The Same Iterated Triple Integral:

The integration is widely used in engineering and mathematics for finding the rectangular area of graphs, we can use the single integral for finding the area of a particular area under the curve, the double integral is used to find the volume of a surface. The triple integral can be used to find the volume like the double integral and also the mass of an object. The integral is useful when the volume of the region is indifferent on X-axis, Y-axis, and Z-axis, and it has variable densities on all three axes. 

We can use the triple integral calculator to find the triple integral of the surface. The utilization of the triple integral is more than the double integral as we can add an extra dimension in the triple integral. When we add an extra dimension of variable density in the same volume, so if the density is variable around three dimensions, we can find the mass of the volume. So by using the double integral, we were only able to find the volume, now by using the triple integral calculation, we are also able to find the mass of the object. Students can use the triple integral calculator to find the integral of an equation:

In this article, we are discussing six ways to write the same iterated triple Integral:


Six ways to write the triple integral :

There are six different ways to write the same iterated triple integral, the function f(x,y,z) remains the same for all the six iterations, the order of the integration, and the limit of the integration change accordingly. For all the iterations the value of the function f(x,y,z) remains the same, we can find the solution of all the iterations by putting the function in the triple integral calculator, this can be quite convenient for the students.


                      With the order of integration x,y,z e f(x,y,z)dxdydz


                     With the order of integration x,z,y e f(x,y,z)dxdzdy


                     With  the order of integration y,x,z e f(x,y,z)dydxdz


                    With  the order of integration y,z,x e f(x,y,z)dydzdx


                      With  the order of integration z,x,y e f(x,y,z)dzdxdy


                      With   the order of integration z,y,x e f(x,y,z)dzdydx


For solving all the iterations of the triple integration, you can use the triple integral calculator and put the limit of the interval in the triple integral calculator for finding the solution of every iteration.


How we solve the triple integral iteration:


We need to find the different limits of each of the above iterations, the limit would change for each of the above-given iterations. We need to use different limits for particular iterations. You need to start to solve the integral from the inside toward the outside integral, for example in solving the integral of dx dy dz, you need to start the integration with respect to “x”, then you have to do the integration with respect to “y”, then with respect to “z”, we can use the cylindrical integral calculator to solve the triple integral of a given function. To evaluate the triple integral we need to start integration from the inside, in this case starting from the “X”.

                  x,y,z e f(x,y,z)dxdydz

Since you are integrating with respect to the “X” and you need to calculate the “Y” and “Z” variables later, so you left the “Y” and “Z” variables, and integrate them according to their respective limits. You solve the x-variable according to the certain limit of the “ X” variable according to its respective limit. You can use the online triple integral calculator according to their respective limits, whether you are solving, with respect to the “X” variable, “Y” variable, or “Z” variable. For solving the triple integral, we can use the triple integral calculator.


             x(y,z)x(y,z) f(x,y,z)dxdydz


Difference between double integral and the triple integral:

When we compare the double integral with the triple integral, we find the triple integral is more useful as compared to the double integral, we can say the double integral is an integration of multivariable functions f(x,y), around a region R, and the interval for the x is (a,b), you can say the limit of the “x” variable is (a,b) and for the “y” variable the limit is (c,d) using vertical e slices of the volume around the surface f(x,y) above the XY-plane or the cartesian coordinates.

Now for the triple integral, we are also integrating multivariable of a function for the density f(x,y,z) for the volume Bintegral, now you can define the limit of each variable, for example for the “x: the interval or the limit is (a,b), for “y” variable the limit is (c,d) and for the “z” variable the limit is (r,s). We are using the changes in the three-dimension plane, which is (x,y,z). Students do find it difficult to solve the triple integral, so you can use the triple integral calculator to find the solution of triple integration.

 







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