Answer to Question #223286 in Calculus for irvon tyuiol

Question #223286

Calculate the volume created by revolution around the axes OX, of D of the plane (OXY) where D={(x,y) ∈R: 0 ≤ y ≤x, (x-2)2 +y2 ≤ 4}



1
Expert's answer
2021-10-26T02:34:31-0400
"V_x=\\pi \\int_a^b f^2(x)dx"

So, we will have



"V_x=V_1+V_2=\\pi \\int_0^2 x^2dx+\\pi\\int_2^4 (\\sqrt{4-(x-2)^2})^2dx"

"\\int_0^2 x^2dx=\\frac{x^3}{3}|_0^2=\\frac{8}{3}"

"\\int_2^4 (4-x^2+4x-4)dx=\\int_2^4 (-x^2+4x)dx=\\newline\n(-\\frac{x^3}{3}+4\\frac{x^2}{2})|_2^4=-\\frac{64}{3}+2\\cdot 16+\\frac{8}{3}-2\\cdot 4=\\newline\n-\\frac{56}{3}+24"

From here

"V_x=(\\frac{8}{3}-\\frac{56}{3}+24)\\pi=(24-\\frac{48}{3})\\pi=(24-16)\\pi=8\\pi" sq.units.


Answer. "8\\pi" sq.units.



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