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

The arc length of the curve is .

Given

The length of the curve y = 4x − 5, −1 ≤ x ≤ 2.

Arc length;

A part of a curve or a part of a circumference of a circle is called Arc. All of them have a curve in their shape.

The given graph y = 4x-5 is the straight line.

The straight line distance between the points is;

Therefore,

The distance between these points is given by;

Hence, the arc length of the curve is .

To know more about arc length click the link given below.

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Ok, here we go.  Pay attention.  The formula for the arc length is .  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 (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 .  Integrating that we have from -1 to 2.  gives us .  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: which simplifies to .  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 .  And there you go!