I need help with the following problem:
Find the exact arc length of the curve y= 2/3 * (x+1)^3/2 where the interval for x is [2,7]. Do the algebra and calculus carefully showing your work. (Final answer should be 38/3).
Here is what I tried to do, but was not able to get a final answer of 38/3:
First I found the derivative: y'=2/3(3/2)(x+1)^1/2 *1 = 1(x+1)^1/2
Then I plugged it into the formula for arc length:
s= ∫[from 2 to 7]Sqrt of 1 + ((x+1)^1/2)^2 dx
s= ∫[from 2 to 7]Sqrt of 1 + x^5/2 + 1^5/2 dx
s= ∫[from 2 to 7]Sqrt of (1+ x^5/2 + 1)^1/2
s= ∫[from 2 to 7]Sqrt of (1+ x^3 + 1)
s= (2 + x^3) | from 2 to 7
s=(2 + 7^3)-(2 + 2^3)
I'm guessing I must have messed up somewhere since my answer isn't anywhere close to 38/3. I am not sure where I messed up or what I did wrong, but if anyone can explain to me how to do this problem or explain what I am doing wrong I would greatly appreciate it. Thanks in advance to anyone who can help.