|SimReal: Mathematics - Multiple Integral - Double Integral - General Area - Ex1 Volume DyDx|
We have the following 3D-object:
z = f(x,y) = 3 - x - y
The area in the xy-plane is a triangle with corners (0,0), (1,0) and (1,1).
In this application we visualize the computation of the volume of this object by double integration.
f(x,y) is the height of this object at position (x,y).
Here this height f(x,y) is 3 - x - y.
In the simulation window you see a tiny bar with area dxdy and height 3 - x - y.
The volume of this tiny bar is f(x,y)dydx or f(x,y)dxdy (green color in the simulation window).
If we add the volume of all such bars over the whole triangle in the xy-plane, we will have the volume of this 3D-object.
The area in the xy-plane is triangular and the limits in the double integral are therefore as shown in the simulation window: y runs from 0 to x (notice it's dependent of x) and x runs from 0 to 1.
By default the integration order is first dydx (you can change to dxdy by the help of the radiobuttons).
Use the scrollbars to visualize the double integration.
By default the integration is first dydx.
Therefore first use the dy scrollbar (the upper one) to show the integration (summation) in the y-direction.
Then use the dx scrollbar (the lower one) to show the integration (summation) in the x-direction.
When you change to integration order dxdy, the scrollbars will automatically change order, so the dx scrollbar to be used first this time will be the upper one, and the dy scrollbar will now be the lower one.
Conclusion: Independent of the choice of dydx or dxdy you should always use the upper scrollbar first.
The 'Move' scrollbar is used to move the shaded area inside the 3D-object to visualize the summation idea of integration.
Using the scrollar (dy or dx) the green color in the inegration expression will (multiplied with dx og dy using the upper scrollbar) show the computed volume of the shaded object.