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Arc Length and Surface Area

Finding the perimeter of arbitrary curves and the area of 3D shapes foils the traditional tools of Geometry, and calls for the help of integrals and derivatives to make these calculations.

Arc Length by Integration

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 $$a$$?

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 $$a+b?$$

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