Sketch the graph of the function, highlighting the part indicated by the given interval,

Question

(a) Sketch the graph of the function, highlighting  the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) Use the integration capabilities of a graphing utility to approximate the arc length.

Answer 1

Step 1:

(a) The function is .

Graph the function by using asymptote.

Vertical asymptotes are zeros of the denominator of a rational function.

Vertical asymptote is .

Since the degree of the numerator is less than the degree of the denominator, horizontal asymptote is .

Graph:

Step 2:

(b)

Apply derivative with respect to .

Definition of the arc length:

If the curve , , then the length of the curve is defined as,

In this case .

Arc length is .

.

Step 3:

(c)

Find arc length using graphing utility:

 Arc length is approximately 2.147 units (using graphing utility).

Solution:

(a)

(b) Arc length is .

(c) Arc length is approximately 2.147 units.