   Chapter 8.1, Problem 1E

Chapter
Section
Textbook Problem

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

To determine

The length of the curve using the arc length formula.

To check: The length of the curve using the distance formula.

Explanation

Given information:

The curve function is y=2x5 (1)

The limits are a=1 and b=3.

Calculation:

The expression to find the length of the curve using the arc length formula is shown below:

L=ab1+(dydx)2dx (2)

Here, the derivative of the function y is dydx, the lower limit is a, and the upper limit is b.

Differentiate Equation (1) with respect to x.

dydx=2

Substitute 2 for dydx, 1 for a, and 3 for b in Equation (2).

L=131+(2)2dx=135dx=5[x]13=5[(3)(1)]

=45

Therefore, the length of the curve is 45_.

Check the length of the curve value by distance formula.

Calculate the value of y for the given x values.

Substitute 1 for x in Equation (1).

y=2(1)5=25=7

Substitute 3 for x in Equation (1)

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