Question

Use the arc length formula to find the length of the curve y = 5x − 1, −3 ≤ x ≤ 2. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.

Answer 1

Answer:

  length = 5√26

Step-by-step explanation:

The arc length of f(x) on the interval [a, b] is given by ...

$\displaystyle d=\int\limits_a^b{\sqrt{1+f'(x)^2}}\,dx$

In the present case, we have ...

  f(x) = 5x -1

  f'(x) = 5

on the interval [a, b] = [-3, 2]. Then the arc length is ...

$\displaystyle d=\int\limits_{-3}^2{\sqrt{1+5^2}}\,dx=\sqrt{26}(2-(-3))\\\\d=5\sqrt{26}$

_____

Check

We're looking for the distance between the points ...

  (-3, f(-3)) = (-3, -16)

and

  (2, f(2)) = (2, 9)

Using the distance formula, the distance is ...

  d = √((2-(-3))² +(9-(-16))²) = √(5² +25²) = 5√(1+5²)

  d = 5√26 . . . . . matches the integral result