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# Answer to Question #184941 in Civil and Environmental Engineering for John Prats

Question #184941

Determine the area of the surface generated by revolving a given curve about a given axis.

a.) y=x³ on (0,1), about the x-axis.

b. ) y=x² on (0,√2), about the y-axis.

1
2021-05-31T03:43:02-0400

a)

"y=x^3"

"\\frac{dy}{dx}=3x^2"

"Surface\\ area\\ of\\ revolution"

"2\\pi\\int_{0}^{1}{x^3\\sqrt{1+\\left(3x^2\\right)^2}\\ dx}"

"let\\ u=1+9x^4"

"du=36x^3\\ dx"

"2\\pi\\int{\\sqrt u\\frac{du}{36}}"

"\\frac{\\pi}{18}\\left(\\frac{2}{3}\\right)\\left(u\\right)^\\frac{3}{2}"

"\\frac{\\pi}{27}\\left(1+9x^4\\right)^\\frac{3}{2}\\ \\ \\ \\ \\ \\left(0,\\ 1\\right)"

"\\frac{\\pi}{27}\\left(\\left(10\\right)^\\frac{3}{2}-1\\right)"

"1.13412\\pi"

b)

"y=x^2"

"x=\\ \\sqrt y"

"\\frac{dx}{dy}=\\frac{1}{2\\sqrt y}"

"Surface\\ are\\ of\\ revolution"

"2\\pi\\int_{0}^{\\sqrt2}{\\sqrt y\\sqrt{1+\\left(\\frac{1}{2\\sqrt y}\\right)^2}\\ dy}"

"2\\pi\\int_{0}^{\\sqrt2}{\\sqrt{y+\\frac{1}{4}}\\ dy}"

"2\\pi\\ \\left(\\frac{2}{3}\\right)\\left(y+\\frac{1}{4}\\right)^\\frac{3}{2}\\ (0,\\ \\sqrt2)"

"\\frac{4\\pi\\ }{3}\\left({\\ \\left(\\sqrt2+\\frac{1}{4}\\right)}^\\frac{3}{2}-\\left(\\frac{1}{4}\\right)^\\frac{3}{2}\\right)"

"2.6958\\pi\\"

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