Finding surface area and constant value of rotated parabolic dish.

Question

An engineer wants to build a parabolic satellite dish. Its shape is formed by rotating the curve y = ax^2 around the y-axis. If the dish must have 4 m of diameter and a maximum depth of 0.5 m, find the value of "a" and the surface area of the dish.

Answer 1

If the diameter is 4, then the x coordinate on either side of the axis of symmetry must be 2. The depth is 1/2 at the points where x = 2 (or -2). So, a(2)^2= 1/2 => 4a = 1/2 => a = 1/8

If y = 1/8 x², the surface is generated by revolving the right half of that parabola from 0 to 2, about the y-axis.

2π∫₀²x√(1+(1/16)x²) dx

Rewrite the radicand by getting LCD, combining the two terms, then taking the square root of the denominator (16)->4. From there u-substitution...

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