Calculus
Arc Length and Surface Area
If \( {f}'(x) = \sqrt{ {x}^4-1 },\) what is the length of the curve \(y = f(x) \) from \(x = 2\) to \(x = 8?\)
What is the length of the curve \(y = \frac{1}{2}( {e}^x+{e}^{-x} ) \) from \(x = -13\) to \(x = 13\)?
Let \(L\) be the length of the line \(y = 3x + 4 \) on the interval \(3 \leq x \leq 8\). If \(L = \sqrt{a}\), then what is the value of \(a\)?
Given \(y \geq 0 , \) what is the length of the curve \({y}^2 = {x}^3 \) on the interval \(0 \leq x \leq 4\)?
Let \(L\) be the length of the curve \(\displaystyle{y = \frac{1}{4}{x}^2 - \frac{1}{2} \ln x }\) for \(4 \leq x \leq 8.\) If \[L = a+\frac{1}{2}\ln{b} ,\] where \(a\) and \(b\) are positive integers, what is the value of \(a+b?\)