Answer to Question #199256 in Calculus for Vincent earl Carmo

Question #199256

Find the centroid of the solid generated if the region bounded by y = 2- x^2, x=0, and y=0 is revolved about the y-axis.


1
Expert's answer
2021-05-30T23:20:46-0400

Solution:

Given, "f(x)=y=2-x^2"




Since, it is revolving around y-axis, centroid will lie on y-axis.

So, it is of the form "C(0,y,0)".

"y=\\dfrac{\\int_0^bf^2(x)dx}{2\\int_a^bf(x)dx}\n\\\\=\\dfrac{\\int_0^{\\sqrt{2}}(2-x^2)^2dx}{2\\int_0^{\\sqrt{2}}(2-x^2)dx}"

"=\\dfrac{\\int_0^{\\sqrt{2}}(4+x^4-4x^2)dx}{2\\int_0^{\\sqrt{2}}(2-x^2)dx}"

"=\\dfrac{(4x+x^5\/5-4x^3\/3)_0^{\\sqrt{2}}}{2(2x-x^3\/3)_0^{\\sqrt{2}}}"

"=\\dfrac{(4{\\sqrt{2}}+({\\sqrt{2}})^5\/5-4({\\sqrt{2}})^3\/3)}{2(2{\\sqrt{2}}-({\\sqrt{2}})^3\/3)}"

"=\\dfrac{(4{\\sqrt{2}}+(4{\\sqrt{2}})\/5-8({\\sqrt{2}})\/3)}{2(2{\\sqrt{2}}-(2{\\sqrt{2}})\/3)}"

"=\\frac45"

Now, "C(0,\\frac45,0)"


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