The area between the y-axis and y= tan x for 0≤ x ≤ pi/4 is rotated about the y-axis.
a) What is the volume of the resulting solid?
b) Suppose the solid is a foam with varying density given by 2/( pi arctan (y)) grams per cubic centimeter, where x and y are given in centimeters. Even though the foam has infinite density y = 0, the total mass is still finite. What is the total mass of the foam?
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Update:the answers are, a) 0.770574 and b) 0.877649... I need to know how they got the answers.. thank you!
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Verified answer
a) Use shells. Each shell has radius r = x
and height h = 1 - y = 1 - tanx, so
V = ∫[a,b] 2πrh dx = 2π ∫[0,π/4] (x - xtanx) dx ≈ 2π(0.122641) = 0.7706 units³
(See cite. Not a pleasant integral; can't be done without limits (that is, as indefinite) unless perhaps you use polynomial expansion.)
b) since y = tanx, density Γ = 2 / (πarctan(tanx)) = 2 / πx, so
mass = ∫[a,b] Γ2πrh dx = 2π ∫[0,π/4] (2/πx)x(1 - tanx) dx = 4 ∫[0,π/4] (1 - tanx) dx
mass = 4(x + ln(cosx)) |[0,π/4] 4(π/4 + ln(√2/2)) ≈ 4(0.4388) = 1.7553 grams
I have exactly twice the answer that you give, but I can't find a mistake.