Question

Use the arc length formula to find the length of the curve y = 4x − 5, −1 ≤ x ≤ 2. check your answer by noting that the curve is a line segment and calculating its length by the distance form

Answer 1

Ok, here we go.  Pay attention.  The formula for the arc length is $AL= \int\limits^b_a { \sqrt{1+( \frac{dy}{dx})^2 } } \, dx$.  That means that to use that formula we have to find the derivative of the function and square it.  Our function is y = 4x-5, so y'=4.  Our formula now, filled in accordingly, is $AL= \int\limits^2_1 { \sqrt{1+4^2} } \, dx$ (that 1 is supposed to be negative; not sure if it is til I post the final answer).  After the simplification we have the integral from -1 to 2 of $\sqrt{17}$.  Integrating that we have $AL= \sqrt{17}x$ from -1 to 2.  $2 \sqrt{17}-(-1 \sqrt{17} )$ gives us $3 \sqrt{17}$.  Now we need to do the distance formula with this.  But we need 2 coordinates for that.  Our bounds are x=-1 and x=2.  We will fill those x values in to the function and solve for y.  When x = -1, y=4(-1)-5 and y = -9.  So the point is (-1, -9).  Doing the same with x = 2, y=4(2)-5 and y = 3.  So the point is (2, 3).  Use those in the distance formula accordingly: $d= \sqrt{(2-(-1))^2+(3-(-9))^2}$ which simplifies to $d= \sqrt{9+144}= \sqrt{153}$.  The square root of 153 can be simplified into the square root of 9*17.  Pulling out the perfect square of 9 as a 3 leaves us with $3 \sqrt{17}$.  And there you go!