Solution:
We are going to find the length of the curve.
Given curve is
"2 y^2 = x^3"
We can write this for y;
"y = \u00b1 \\frac {1} {\\sqrt 2} x^{\\frac {3}{2}}"Differentiate with respect to x,
"= \u00b1 \\frac {1} {\\sqrt 2} \\times \\frac {3} {2} \\times x^{\\frac {1}{2}} == \u00b1 \\frac {3} {2\\sqrt 2} \\times x^{\\frac {1}{2}}"
We can find the length of the given curve "2 y^2 = x^3" from (0, 0) to the point (4,4√2) by using a formula.
That is,
Length of the curve =
"=\\int_0^ 4 \\sqrt {(1+ (\\frac {dy}{dx})^2 } \\space dx""= \\int_0^ 4 \\sqrt {(1+ (\u00b1 \\frac {3} {2\\sqrt 2} \\times x^{\\frac {1}{2}})^2 } \\space dx"
"= \\int_0^ 4 \\sqrt {(1+ ( \\frac {9} {8} \\times x) } \\space dx"
"= \\frac {1} {2\\sqrt 2 } \\int_0^4 \\sqrt {8+ {9} x } \\space dx"
"Let \\space 8 +9x = t\\\\ \n 9 dx = dt \\\\\n dx = \\frac {1}{9} dt"
"Limits : x = 0 \\space then \\space t = 8 \\\\\n x = 4 \\space then \\space t = 44"
"= \\frac {1} {2\\sqrt 2 } \\int_8^{44} \\sqrt t \\space \\frac {1}{9} \\space dt"
"= \\frac {1} {18\\sqrt 2 } \\int_8^{44} t^{\\frac {1}{2}} \\space \\space dt"
"= \\frac {1} {18\\sqrt 2 } \\space [\\frac {t^{\\frac {1}{2} + 1}} {\\frac {1}{2} +1}]_8^{44}"
"= \\frac {1} {18\\sqrt 2 } \\space \\frac {2}{3} ((44)^{\\frac {3}{2}} - (8)^\\frac{3}{2})"
"= \\frac {1} {27\\sqrt 2 } (88\\sqrt {11} - 16 \\sqrt 2)"
Answer:
"Length \\space of \\space the \\space curve= \\frac {1} {27\\sqrt 2 } (88\\sqrt {11} - 16 \\sqrt 2)"
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