Question

Draw the curve for the function \(y = x + \cos x\)and find the length of the curve.

Short answer

The length of the curve by the use of the graph is 1.7. The length of the curve by the use of calculator is 1.7294.

Step-by-step solution

Given data

The curve function is \(y = x + \cos x\). The lower limit is \(a = 0\) and the upper limit is \(b = \frac{\pi }{2}\).

Concept of Arc length

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

Differentiate the equation

Draw the graph for the function \(y = x + \cos x\) as shown in Figure 1.

The length of the curve is slightly larger than the hypotenuse of the triangle.

The triangle is formed by the co-ordinate \((1,1),\left( {\frac{\pi }{2},1} \right)\), and \(\left( {\frac{\pi }{2},\frac{\pi }{2}} \right)\).

Calculate the length of hypotenuse as follows: \(\sqrt {{{\left( {\frac{\pi }{2}} \right)}^2} + {{\left( {\frac{\pi }{2} - 1} \right)}^2}} \approx 1.7\)

Therefore, the length of the curve from the graph is 1.7

The expression to find the length of the curve \((L)\) is shown below:

\(y = x + \cos x\) ……. (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 \(x\): \(\frac{{dy}}{{dx}} = 1 - \sin x\).

Substitute the values

Substitute \(1 - \sin x\) for \(\frac{{dy}}{{dx}},0\) for \(a\), and \(\frac{\pi }{2}\) for \(b\) in Equation (2).

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

Solve Equation (3) by the use of calculator; \(L \approx 1.7294\).

Therefore, the length of the curve by the use of calculator is \(1.7294\).