Question

Determine and represent the integral as the length of the curve.

Short answer

The length of the curve is \(3.8202\).

Step-by-step solution

Given data

The curve function is \(y = \sin x\).

The limits are \(a = 0\) and \(b = \pi \).

Concept of Arc length

Arc length is the distance between two points along a section of a curve.

Differentiate the equation

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

\(y = \sin x\) ….... (1)

\(L = \int_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} } dx\) …… (2)

Here, the derivative of the function \(y\) is \(\frac{{dy}}{{dx}}\), the lower limit is \(a\), and the upper limit is \(b\).

Differentiate Equation (1) with respect to \(X\):

\(\frac{{dy}}{{dx}} = \cos x\)

Integrate the equation

Substitute \(\cos x\) for \(\frac{{dy}}{{dx}}\) in Equation (2).

\(L = \int_0^\pi {\sqrt {1 + {{\cos }^2}x} } dx\) …….. (3)

Equation (3) represents the length of the curve.

Solve Equation (3) and use the calculator.

\(L \approx 3.8202\)

Therefore, the length of the curve is \(3.8202\).