Volumes of solids using calculus

[b^3]

If you're only taking the first and 4th quadrants then yes, it will produce the top of a circus tent, but b^3 has taken all 4 quadrants, so his would be like from x = -3 to x = 3, what you're saying is from x = 0 to x = 3

Wait, no I've only taken the first quadrant, the area that is between the axis and y=x^2 and y=4. I think its just a bad angle, the backside of that shape is flat (corresponding to x=0), the circus tent shape is the shape that is cut out of that shape.

[darkmaster25]

but the parabola example is different because the area enclosed between the parabola and x axis is different to the area enclosed between the parabola and the y axis. here the area regardless of on which axis is rotated, is the same.

[b^3]

OK before we go off here, note that the one that is being rotated around the x-axis, is still the region that is enclosed with the y-axis (and the line y=4), the reason I did this is to keep the areas/regions the same so that I could show that the shape when rotated about the different axis is different and will result in a different volume, even though we started with the same region.

It would be different if it were the region of enclosed between the graph, the x-axis, and another line.

I hope that makes sense.

but the parabola example is different because the area enclosed between the parabola and x axis is different to the area enclosed between the parabola and the y axis. here the area regardless of on which axis is rotated, is the same.

That is basically what I just said....

Just because the area in the region is the same, does not mean that the volume from the two shapes is the same as the distance from axis to the outside of the region is larger or smaller depending on which axis you rotate it on, so you get two different shapes and areas when you rotate the shape on the different axis even if the areas in the reigon is the same.

[darkmaster25]

OOO i get it!! finally! it's just visually hard to picture but i can see what you mean b^3. the radius of the one going around the y axis will be bigger than the one compared to the one going around the x-axis. I can't believe i didn't catch that. 

Thanx..     

[b^3]

I probably should add that the interval of integration is also different. The point is that you are dealing with a completely different set of thin slices of the 3-D shape.