... Er, I think what I really want is , but anyway.
So I'm asked to evaluate the integral by changing to spherical coordinates, .
I see how do to do all the plugging in, but I'm not sure how to determine the limits of integration. I know my radius is going to start at 0, but I want it to be limited by some function of . I can see that I want to be looking only at what's above the xy-plane and toward the positive side of x, so I want and . So I guess all that I don't see is what function bounds the radius on top.
No, did you sketch the region? you're integrating over the volume of part of an ice cream cone, the bottom part comes from the cone , the top part comes from the sphere . You're in the first quadrant only.
You have
, and
Draw the figure and see if you can figure these out. begin with the traces
Oh, right, my x is positive and y positive and z positive, so I'm just looking at the first octant, hence . Because it's a cone and when keeping y fixed so it's a 45-degree angle hence . And because the top part is a portion of a circle the radius reaches out to a constant end-point, in this case when we plug in for the upper limit. Hopefully that's the right way to think of it.
yup
you mean if you put y = 0 you get that. I suppose you can think of it that way, but it seems somewhat limited. Notice that if you view the figure sideways (positive z-axis pointing up, the xy-plane is the horizontal), a part of the figure will be a right triangle whose height and base lengths are z. this is an isosceles right triangle, and hence the other two angles are , which gives you yourBecause it's a cone and when keeping y fixed so it's a 45-degree angle hence .
is the distance from the origin to the outer shell of the solid figure, which in this case is the distance from the origin to the sphere. the distance is therefore the radius of the sphere, which is , and that gives you yourAnd because the top part is a portion of a circle the radius reaches out to a constant end-point, in this case when we plug in for the upper limit. Hopefully that's the right way to think of it.
from these you can find the limits. the moral? if possible, draw the region they are talking about, usually you can use basic geometry/trigonometry to figure out the limits.