Use polar coordinates to find the volume of the given solid. Bounded by the para

Question

Use polar coordinates to find the volume of the given solid. Bounded by the paraboloid z = 6 + 2x2 + 2y2 and the plane z = 12 in the first octant

Explanation / Answer

Since 1st quadrant means x>0 and y>0 in 2D space, I guess 1st octant means x>0, y>0 and z>0 in 3d space. The way I would do this is add up slices. i.e. at any particular z value, work out the area A of the slice. i.e. V = (integrate z from 0 to 8) A * dz At any z, the formula for the cross section plane is: 2x^2 + 2y^2 = z - 2 (OK, so notice this means that the solid only exists for z>=2) => x^2 + y^2 = z/2 - 1 So that's a circle with r^2 = (z/2 - 1) => area of circle = pi r^2 = pi(z/2 - 1) But we only want the part of the circle which is in the first quadrant. Luckily, this circle is centred at (0,0), so area in the 1st quadrant is just 1/4 of that. i.e. Area A = pi(z/2 - 1)/4 = pi(z - 2)/8 And now V = (integrate z from 2 to 8) A dz etc.

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