Finding phi1: z=sqrt (x^2+y^2). When y=0 the equation becomes z=x, with slope 1. This slope has the angle from the vertical of pi/4, so phi1=pi/4. For z= sqrt (3x^2+3y^2). y=0 -> z=sqrt(3)x, slope=sqrt(3). Slope is tan(angle from horizontal) and slope is cot(angle from vertical), so phi1 = arctan(1/sqrt(3)) = pi/6 or phi = pi/2-arctan(sqrt(3)) = pi/6. The angle of the side of the cone, phi, is quite literally sitting right there in the coefficient in front of the sqrt(x^2+y^2). Just invert the coefficient and take the arctan.
@cchamberlin20002 жыл бұрын
True. I mention that in reply to one of the older questions.
@sbusisomcebongwenya167 Жыл бұрын
Well. Im saving this one. Thank you sir!!!
@rashedbinkhonain70223 жыл бұрын
Great video!
@Srijjjj3 жыл бұрын
Good one sir
@sandlertossone18136 жыл бұрын
Won't a cone always hit the sphere at pi/4?
@calculus3d1306 жыл бұрын
No. Use z=sqrt(3x^2+3y^2) for your half-cone. You should get phi=pi/6
@calculus3d1306 жыл бұрын
Here's another way to get phi, assuming z=sqrt(3x^2+3y^2). Set y=0, which turns the side of the cone into a line in the xz-plane. You'll get z=sqrt(3)*x. Draw the corresponding triangle and get tan(phi)=1/sqrt(3) or phi=pi/6.
@nathannixon27276 жыл бұрын
What if the cone's summit isint centered with the sphere?
@calculus3d1306 жыл бұрын
@@nathannixon2727 If the cone's vertex doesn't coincide with the center of the sphere, then it is potentially a harder problem for which spherical coordinates may not be helpful. Still, in general, to find the volume of a solid region, you would triple integrate: the inner lower limit of integration would be the function that gives the lower boundary surface, while the inner upper limit would be the function that represents the upper boundary surface.
@bangaloremathematicalinsti53514 жыл бұрын
what is the software you are using to draw the graphs?
@calculus3d1304 жыл бұрын
OneNote and a cheap Wacom drawing tablet.
@lol-gu4wr7 ай бұрын
nice video
@MrSqueakinator4 жыл бұрын
Don't you need to be integrating z for your first integral?
@calculus3d1304 жыл бұрын
You could do it that way in rectangular or cylindrical coordinates. I used spherical coordinates here, so no need to worry about z.
@KishoreG23965 жыл бұрын
I dont understand. If you rewrite the cone equation with z = pcos(phi), x = psin(phi)cos(theta), and y = psin(phi)sin(theta), you will get sin(phi) = cos(phi), which implies phi = pi/4, not pi/6. Could you please explain?
@calculus3d1305 жыл бұрын
In the example from the video, the lower value of phi does equal pi/4. If you're referring to the example z=sqrt(3x^2+3y^2), then, no, phi must be smaller than pi/4 because this cone is "taller" than the one from the video, given by z=sqrt(x^2+y^2). To see this, set y=0 in z=sqrt(3x^2+3y^2). This turns the side of the cone into a line in the xz-plane. You'll get z=sqrt(3)*x. Draw the corresponding triangle and get tan(phi)=1/sqrt(3) or phi=pi/6. Remember, the lower limit on phi in this case is measured from the z-axis to the side of the cone (or line if you look at the two-dimensional cross-section you get by setting y=0).
@KishoreG23965 жыл бұрын
@@calculus3d130 I understand now. Thank you.
@calculus3d1305 жыл бұрын
@@KishoreG2396 Glad to help.
@jumanamohamediqbal65313 жыл бұрын
Isn't it 0 to π/4
@calculus3d1303 жыл бұрын
No, assuming you're referring to the limits on phi, that would give the volume inside the cone and the sphere.