Question

Find the exact length of the curve
y=2/3(x^2-1)^(3/2),  1 ≤ x ≤ 3
 i dont know ho to start off can someone help me

Accepted Answer

Notice that the curve whose length you should find is defined by a rectangular equation, hence, you need to use the next formula such that:
int_a^b sqrt(1 + ((dy)/(dx))^2)dx
You need to differentiate the given curve using the chain rule such that:
((dy)/(dx)) = (2/3)*(3/2)*(x^2 - 1)^(3/2 - 1)*(x^2 - 1)'
((dy)/(dx)) = 2x*sqrt(x^2 - 1)
You need to raise to square such that:
((dy)/(dx))^2 = 4x^2*(x^2 - 1)
You need to add 1 such that:
((dy)/(dx))^2 + 1 = 4x^4 - 4x^2 + 1
((dy)/(dx))^2 + 1 = (2x^2 - 1)^2
Taking the square root both sides yields:
sqrt(1 + ((dy)/(dx))^2) = sqrt((2x^2 - 1)^2)
sqrt(1 + ((dy)/(dx))^2) = |2x^2 - 1|
You may evaluate the length of the curve such that:
int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = int_1^3 (2x^2 - 1)dx
int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = int_1^3 (2x^2)dx - int_1^3 dx
int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = (2x^3/3 - x)|_1^3
int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = 54/3 - 3 - 2/3 + 1
int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = 52/3 - 2 
int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = 46/3
Hence, evaluating the length of the given curve yields ds = int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = 46/3 .

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Find the exact length of the curve y=2/3(x^2-1)^(3/2), 1 ≤ x ≤ 3 i dont know ho to start off can someone help me

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