Set up a triple integral for the volme of the solid. I dont understand how you f

Question

Set up a triple integral for the volme of the solid.

I dont understand how you figure out the x bounds from the given data. Also does it matter what order it is in? Would the answer still be correct if I wrote it in dzdxdy order?

Set up a triple integral for the volme of the solid. The solid bounded by z=9- I dont understand how you figure out the x bounds from the given data. Also does it matter what order it is in? Would the answer still be correct if I wrote it in dzdxdy order? , z=0 ,y=0 ,y=2x The answer is

Explanation / Answer

The bounds indicate the maximum and minimum value.

Here for checking the area covered on the x axis, we equate the equations having x to 0.

That is:

9-x2=0 and 2x=0

So the values are x=3 and x=0

Thus we get the bounds

And coming to the order, it definitely matters until the respective integral also changes the place.

I.e. now we see the that the integrals are in the order of x,y,z and the operators in the order dz,dy,dx

Here the integral and operator act as a set. here z goes with dz,y with dy and x with dx.

Hence if we change the place of operator then the integral also moves in opposite direction.

For example if dy moves to the right, integal bounds of y move to left and so on.

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