Answer to Question #100153 in Calculus for Joseph Yep

Question #100153
1. Let L be the length of the curve y= x^5 + 8/ 16x^2 from x=2 to x=3, an integral expression for L is ________.

2. Let L be the length of the curve x= y^4/8 + 1/ 4y^2 from y= 1 to y=4, an integral expression for L is______.
1
Expert's answer
2019-12-09T12:38:32-0500

Length of the curve is "\\int\\limits_{t_1}^{t_2}\\sqrt{(x'(t))^2+(y'(t))^2}dt" , where "(x(t),y(t)), t\\in[t_1,t_2]" is parametrization of the curve.

1)We have "x(t)=t" and "y(t)=\\frac{t^5+8}{16t^2}" , "t\\in[2,3]"

So "x'(t)=1" and "y'(t)=\\frac{5t^4\\cdot 16t^2-(t^5+8)\\cdot 32t}{256t^4}=\\frac{3t^5-16}{16t^3}"

"L=\\int\\limits_2^3\\sqrt{1+\\bigl(\\frac{3t^5-16}{16t^3}\\bigr)^2}dt"

2)We have "x(t)=\\frac{t^4}{8}+\\frac{1}{4t^2}" and "y(t)=t" , "t\\in[1,4]"

"x'(t)=\\frac{t^3}{2}-\\frac{1}{2t^3}=\\frac{t^6-1}{2t^3}", "y'(t)=1"

"L=\\int\\limits_1^4\\sqrt{1+\\bigl(\\frac{t^6-1}{2t^3}\\bigr)^2}dt=\\int\\limits_1^4\\frac{t^6+1}{2t^3}dt"

Answer: 1)"\\int\\limits_2^3\\sqrt{1+\\bigl(\\frac{3t^5-16}{16t^3}\\bigr)^2}dt" , 2)"\\int\\limits_1^4\\frac{t^6+1}{2t^3}dt"


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