Answer to Question #124050 in Calculus for Desmond

Question #124050
Find the volume of the solid generated by revolving the region bounded by the curves
y = x² and y = 4x - x² about the line y = 6.
1
Expert's answer
2020-06-29T16:24:36-0400

"y=x^{2} \\quad ,y=4 x-x^{2} \\quad ,y=6 \\\\[1 em]\n\\therefore x^{2}=4 x-x^{2} \\rightarrow x^{2}-2 x=0 \\rightarrow x=0,2 \\\\[1 em]\n\\therefore V=\\pi \\int_{0}^{2} r_{1}^{2}-r_{2}^{2} d x \\\\[1 em]\nr_{1}=6-x^{2} \\quad, r_{2}=6-4 x+x^{2} \\\\[1 em]\n\n\n\\therefore V=\\pi \\int_{0}^{2}\\left(6-x^{2}\\right)^{2}-\\left(6-4 x+x^{2}\\right)^{2} dx\\\\[1 em]\n=\\pi \\int_{0}^{2} 36-12 x^{2}+x^{4}-\\left(x^{4}-8 x^{3}+28 x^{2}-48 x+36\\right) d x \\\\[1 em]\n=\\pi \\int_{0}^{2} 36-12 x^{2}+x^{4}-x^{4}+8 x^{3}-28 x^{2}+48 x-36 d x \\\\[1 em]\n=\\pi \\int_{0}^{2} 8 x^{3}-40 x^{2}+48 x d x \\\\[1 em]\n=\\pi\\left(2 x^{4}-\\frac{40}{3} x^{3}+24 x^{2}\\right) \\bigg|_{0}^{2} \\\\[1 em]\n\\therefore V\\approx 67"


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