Question

In Exercises \(15-22,\) (a) graph 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 are length. $$ y=\sin x, \quad 0 \leq x \leq \pi $$

Short answer

The definite integral representing the arc length of the curve \(y=\sin x\) in the interval \(0 \leq x \leq \pi\) is \[L = \int_0^\pi\sqrt{1+ (\cos x )^2} dx\]. Due to the complexity of the integral, it cannot be evaluated using the typical elementary functions and techniques. A numerical approximation can be achieved using the integration capabilities of a graphing utility. The actual numerical approximation would depend on the specific graphing utility used.

Step-by-step solution

Graph the Function

To graph the function \(y=\sin x\) in the interval \(0 \leq x \leq \pi\), simply plot the sine curve starting from x=0 till x=\(\pi\). The sine curve starts at 0, rises to 1 at x=\(\pi/2\) and then falls back down to 0 at \(x=\pi\).

Definite Integral for Arc Length

The formula to find the arc length of a curve defined by a function \(f(x)\) from \(a\) to \(b\) is: \[L = \int_a^b \sqrt{1 + [f'(x)]^2} dx\]\n\nTo use this formula for the given function \(y=\sin x\), we first need to derive it. The derivative of \(y=\sin x\) is \(y'=\cos x\). Substituting \(f'(x)=\cos x\) into the formula, we get the definite integral \[L = \int_0^\pi \sqrt{1+ (\cos x )^2} dx \]\n\nThis integral cannot be evaluated using elementary functions, and thus, we can't solve it using the techniques studied so far.

Approximate the Arc Length

To approximate the arc length, we can use the integration capabilities of a graphing utility. The graphing calculator will compute a numerical approximation of the definite integral. The exact process might vary depending on the specific graphing calculator used, but generally, you would enter the function under 'Y=', go to the 'CALC' menu, select '7: Integral' and then input the bounds 0 and \(\pi\).

Key concepts

Graphing Sine Function

Understanding how to graph the sine function is foundational in trigonometry and calculus. When graphing the basic sine function, denoted as \(y = \text{sin} x\), the pattern it follows is a continuous wave that oscillates between a maximum value of \(1\) and a minimum value of \(-1\). For the interval \(0 \text{leq} x \text{leq} \text{pi}\), the function starts at \(0\), increases to its peak at \(x = \text{pi}/2\), where \(y = 1\), and descends back to \(0\) at \(x = \text{pi}\).

The sine curve portrays periodic behavior, meaning it repeats this pattern at regular intervals, with a period of \(2\text{pi}\) radians. Recognizing this wavy pattern and understanding its attributes, such as amplitude and period, are crucial for sketching accurate graphs of sinusoidal functions. When graphing, one helpful tip is to mark known values or 'key points' along the axis, such as where the function crosses the axis, reaches a maximum, or reaches a minimum.

Definite Integral

The concept of a definite integral is central to calculus, particularly when measuring the arc length of a curve. A definite integral represents the accumulated sum of infinitesimally small quantities over a given interval. It can be thought of as the area under a curve between two points on the x-axis.

For the arc length, the specific integral used is \[L = \text{int}_a^b \text{sqrt}{1 + [f'(x)]^2} dx\], where \(f'(x)\) is the first derivative of the curve's function. This formula encapsulates the Pythagorean theorem's infinitesimal application to measure length. When applied to the sine function, with its derivative being \(y' = \text{cos} x\), we substitute to find the arc length integral for \(y = \text{sin} x\) over the interval \([0, \text{pi}]\). The resulting integral incorporates the square of the cosine function and cannot be resolved with basic calculus techniques, indicating the need for numerical approximation methods.

Numerical Approximation

Numerical approximation methods come into play when an exact answer is either impossible or impractical to obtain. Such is the case with complex definite integrals like the one for the arc length of the sine curve. Approximation techniques, such as Riemann sums, trapezoidal rule, or Simpson's rule, estimate the value by summing the areas of shapes (like rectangles or trapezoids) that approximate the region under the curve.

In modern calculus practice, a graphing utility or calculator can employ these methods to compute the integral to a high degree of accuracy. The process typically involves entering the function into the calculator and using its built-in integration function to estimate the integral's value between the given bounds. Modern graphing utilities offer a means to overcome the limitations of analytical integration, providing students with tangible solutions to complex problems.