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Triple Integral Calculator (with competition)

Krystian Karczyński

Founder and General Manager of eTrapez.

Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.

He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.


Some time ago the irreplaceable Wolfram shared with us his calculator for triple integrals (I translated it a bit):

As you can see above, the calculator is very simple. In the first line, we enter the integrand and the order of integration. Click ‘Count’. We got the result 🙂

Function symbols in Tungsten

Markings can be a bit of a problem, e.g. square root of 4 minus x squared end root within Wolfram’s integration limits, you would have to enter as ‘sqrt(4-x^2)’ – as in the example above. Take a look at the General Instructions for Typing Mathematical Formulas and if you have any problems, feel free to ask in the comments.

The calculator perfectly copes with calculations without cylindrical, spherical and numbers coordinatesp.me – in one word HONEY.

Example 1

Using a calculator, we will calculate the triple integral of the functionf open parentheses x point y point z close parentheses equals x y squared z minus 1 in a ball with an equationx squared plus open parentheses y minus 1 close parentheses squared plus z squared equals 1 .

After drawing, the integration area would look like this (unfortunately, the calculator will not help us with this):

Sphere centered at (0,1,0)

With a calculator at hand, we don’t even enter spherical coordinates, we’re such lazy people. We describe the area of ​​integration normally with x,y,z coordinates.

The projection of the ball onto the plane (say) xOy will be a circle:

Sphere projection on xOy plane (circle)

In this circle the ‘x’s – let’s say again – let them vary within a constant range of -1 to 1.

Let the ‘y’s change within the limits of variables, we calculate them from the equation of the sphere taking awith equals 0 :

x squared plus open parentheses y minus 1 close parentheses squared plus z squared equals 1

x squared plus open parentheses y minus 1 close parentheses squared plus 0 squared equals 1

x squared plus open parentheses y minus 1 close parentheses squared equals 1

open parentheses y minus 1 close parentheses squared equals 1 minus x squared

y minus 1 equals square root of 1 minus x squared end root or y minus 1 equals negative square root of 1 minus x squared end root

y equals square root of 1 minus x squared end root plus 1 or y equals negative square root of 1 minus x squared end root plus 1

And we have exactly the limits of integration for y. Integration limits for z are calculated (these will be surfaces) similarly from the equation:

x squared plus open parentheses y minus 1 close parentheses squared plus z squared equals 1

z squared equals 1 minus x squared minus open parentheses y minus 1 close parentheses squared

z equals square root of 1 minus x squared minus open parentheses y minus 1 close parentheses squared end root or z equals negative square root of 1 minus x squared minus open parentheses y minus 1 close parentheses squared end root

We do not care about cleaning at all, because we have a calculator.

To sum up:

We have an integrand function: f open parentheses x point y point z close parentheses equals x y squared z minus 1

We have the area of ​​integration:

Integration limits

Entering data into the calculator

We enter the integrand and the limits of integration into the calculator as follows:

Calculator with filled fields

The order of integration agrees with us (first after ‘z’, then after ‘y’, and finally in constants after ‘x’), so we only need to click on ‘Count’  and we have the result:

Calculator result

So we have the result: -4.18879, calculated neatly and without spherical coordinates.

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