An exact length of arc can be computed with the arc length formula, together with a bit of algebra, so the integrand is a product of factors where one factor is the derivative of the other so we can use "U-substition"

  An exact length of arc can be computed with the arc length formula, together with a bit of factoring, recalling a^2+2ab+b^2 = (a + b)^2

  Knowing how to use calculus to find arc length helps us find the surface area a function rotated around an axis.  I use u-substitution to find the surface area of y=x^2 two different ways.

Use limits to test Continuity! We check if the function exits, the limit from left and right exist, and they are all match! 

 A function can be continuous at a point, and interval or everywhere.  We can tell if three things are required!

 We can use Theorems together with algebra tricks to find limits that have the indeterminate 0/0 form.

To find the limit, we turn to the conjugate once more!

Don't fear the "∆x" it works like a single variable, and refers to the change in x.

 Her we take advantage of (a+b)(a-b) = a2 - b 2. to resolve an indeterminate form.

 If you substitute 4 for x, we get a 0/0 situation we call an indeterminate form which can resolved with some algebra tricks.