Question

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

Short answer

The answer was double-checked by noticing that the curve is a line segment and arc length is\({\rm{4}}\sqrt {\rm{5}} \).

Step-by-step solution

To find arclength.

\(\begin{aligned}{}{\rm{y = 2x - 5}}\;\;\;{\rm{\{ - 1}} \le {\rm{x}} \le {\rm{3\} }}\\\frac{{{\rm{dy}}}}{{{\rm{dx}}}}{\rm{ = }}\frac{{\rm{d}}}{{{\rm{dx}}}}{\rm{(2x - 5)}}\\{\rm{ = 2 - 0}}\\{\rm{ = 2}}\end{aligned}\)

Apply the arc length formula\({\rm{L = }}\int_{\rm{a}}^{\rm{b}} {\sqrt {{\rm{1 + }}{{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)}^{\rm{2}}}} } {\rm{dx}}\) now.

Here\({\rm{a = - 1}}\),\({\rm{b = 3}}\)

\(\begin{aligned}{}{\rm{L = }}\int_{{\rm{ - 1}}}^{\rm{3}} {\sqrt {{\rm{1 + (2}}{{\rm{)}}^{\rm{2}}}} } {\rm{dx}}\\{\rm{L = }}\int_{{\rm{ - 1}}}^{\rm{3}} {\sqrt {\rm{5}} } {\rm{dx}}\\{\rm{\;L = }}\sqrt {\rm{5}} {\rm{ \times (x)}}_{{\rm{ - 1}}}^{\rm{3}}{\rm{\;}}\\{\rm{L = }}\sqrt {\rm{5}} {\rm{ \times (3 - ( - 1))}}\\{\rm{L = }}\sqrt {\rm{5}} {\rm{ \times }}\left( {\rm{4}} \right)\\{\rm{L = 4}}\sqrt {\rm{5}} \end{aligned}\)

Check.

A straight line is the provided equation. Using the distance formula, one may calculate the distance between two points.

\({\rm{d = }}\sqrt {{{\left( {{{\rm{x}}_{\rm{2}}}{\rm{ - }}{{\rm{x}}_{\rm{1}}}} \right)}^{\rm{2}}}{\rm{ + }}\left( {{{\rm{y}}_{\rm{2}}}{\rm{ - }}{{\rm{y}}_{\rm{1}}}} \right)} \)

Find the equivalent value of \({\rm{y}}\)coordinates given \({{\rm{x}}_{\rm{1}}}{\rm{ = - 1}}\)and\({{\rm{x}}_{\rm{2}}}{\rm{ = 3}}\).

\(\begin{aligned}{}{{\rm{y}}_{\rm{1}}}{\rm{ = 2}} \cdot {{\rm{x}}_{\rm{1}}}{\rm{ - 5 = 2}} \cdot {\rm{ - 1 - 5 = - 7}}\\{{\rm{y}}_{\rm{1}}}{\rm{ = 2}} \cdot {{\rm{x}}_{\rm{2}}}{\rm{ - 5 = 2}} \cdot {\rm{3 - 5 = 1}}\end{aligned}\)

Length of arc

\(\begin{aligned}{}{\rm{ = }}\sqrt {{{{\rm{(3 - ( - 1))}}}^{\rm{2}}}{\rm{ + (1 - ( - 7)}}{{\rm{)}}^{\rm{2}}}} \\{\rm{ = }}\sqrt {{{{\rm{(3 + 1)}}}^{\rm{2}}}{\rm{ + (1 + 7}}{{\rm{)}}^{\rm{2}}}} \\{\rm{ = }}\sqrt {{\rm{16 + 64}}} \\{\rm{ = 4}}\sqrt {\rm{5}} \end{aligned}\)

Therefore, the answer was double-checked by noticing that the curve is a line segment and arc length is\({\rm{4}}\sqrt {\rm{5}} \).