A satellite receiver dish for a network like DirecTV is about 50cm across and 25

Question

A satellite receiver dish for a network like DirecTV is about 50cm across and 25 cm deep.  It's not a perfect parabola shape, but we will approximate it with a nice parabola.  Suppose it's defined by rotating the region between the following functions around the y-axis (all units in cm): front surface: y=x^2 / 25 + 1 back surface:  y=x^2 / 25 - 1 for x from 0 to +25.  a) Compute the volume of the plastic that makes up the dish by doing an integration. b) Compare to the volume of a flat plastic dish that has a radius of 25cm and a thickness of 2cm.  Comment as appropriate. c) It turns out after actually measuring a dish (rather than imagining one), the depth is only 4 cm, not 25.  Figure out the new equations for the front and back surfaces, and recompute the volume as in part (a).

Explanation / Answer

a) 2pi int x [(x^2/25 +1)- (x^2/25-1)] dx from x=0 to x=25

2pi int 2x dx= 2pi x^2 from 0 to 25= 1250 pi

b) Flat plastic disk would be pi r^2 h= pi 25^2 *2= 1250pi (Volumes are same).

c) 2pi int [x(x^2/4 +1) -(x^2/4 -1)] dx from x=0 to x=4

2pi int 2x dx from x=0 to 4 would be 2pix^2 = 32pi

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