Double integrals question.

bllnsr said:
Find the volume of the solid in the first octant bounded by the coordinates planes, the plane x=3 and the parabolic cylinder z = 4-(y^2).

Its to confusing for me I am having difficult time solving these questions.
here is what I sketched.

http://img818.imageshack.us/img818/2...0530211915.jpg
Click to expand...


The "parabolic cylinder" given is kind of like an upside-down half pipe, or a vine archway like this:

flat,550x550,075,f.jpg


In fact let's use this picture (all rights reserved to the owner, yada yada). The x-axis is the road going through the middle so the xz plane is cutting it in half, down the middle. The xy-plane is the ground. The start of the archway (where the brick begins) might be the yz plane, and where the archway ends, the plane x=3.

Now, it doesn't matter which side we reference to be the "positive" y side of the archway, but let us use the right half of the archway. Then your solid is just half of the space inside, to the right of the road under the archway. The archway has a "height" of four, going down to zero at y=2, where the archway touches the ground.
 
bllnsr said:
Find the volume of the solid in the first octant bounded by the coordinates planes, the plane x=3 and the parabolic cylinder z = 4-(y^2).

Its to confusing for me I am having difficult time solving these questions.
here is what I sketched.

http://img818.imageshack.us/img818/2...0530211915.jpg
Click to expand...
Your sketch may not be as beautiful as daon2's - but should be adequate!

Integration in the x direction is just multiplication by 3.

Can you find the area of half the arch, by integrating y*z(y) dy over appropriate limits?
 
Finding limits is what is troubling me.

I did quite a lot of searching on google and sketched the graph again.

http://img826.imageshack.us/img826/7167/cylin.jpg

So what I got from it is that for 'y=0' 'z' will be 4 and for 'y=2' 'z' will be zero so limits of y will be 0 to 2.
We can't go any further because for any values of y greater than 2 it will go into the negative region of x-axis right?
Also will the limits of x be from 0 to 3?
 
bllnsr said:
Finding limits is what is troubling me.

I did quite a lot of searching on google and sketched the graph again.

http://img826.imageshack.us/img826/7167/cylin.jpg

So what I got from it is that for 'y=0' 'z' will be 4 and for 'y=2' 'z' will be zero so limits of y will be 0 to 2.
We can't go any further because for any values of y greater than 2 it will go into the negative region of x-axis right?
Also will the limits of x be from 0 to 3?
Click to expand...
-x and y seem to be reversed in your picture.

Your limits are correct: 0 < y < 2,.....0 < x < 3

An increment of Volume is.... dV = z(y) * dy * dx = (4 - y^2) dy dx

Can you proceed from there?
 
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