Problem 33 - Calculus Early Transcendentals

Question

Average height of an arch The height of an arch above the ground is given by the function \(y=10 \sin x,\) for \(0 \leq x \leq \pi\). What is the average height of the arch above the ground?

Step-by-Step Solution

Answer

Based on the given function of the arch's height, \(y=10 \sin x\), and the interval \(0 \leq x \leq \pi\), we found the integral of the function within this interval and divided it by the length of the interval. As a result, the average height of the arch above the ground is \(\frac{10}{\pi}\) units.

Step 1: Write down the given function and interval

We are given the height function of the arch as \(y=10 \sin x\), and the interval we are working with is \(0 \leq x \leq \pi\).

Step 2: Find the integral of the function within the given interval

We need to calculate the integral of the function within the interval. Integrate the function \(y=10 \sin x\) with respect to \(x\) from \(0\) to \(\pi\): $$ \int_{0}^{\pi} 10 \sin x \, dx $$

Step 3: Calculate the integral

Applying the integral rule for \(\sin\) function: $$ \int \sin x \, dx = -\cos x + C $$ So, the integral of the height function becomes: $$ \int_{0}^{\pi} 10 \sin x \, dx = \left[ -10\cos x \right]_{0}^{\pi} $$ Evaluate the integral at the bounds \(0\) and \(\pi\): $$ \left[ -10\cos(\pi) - (-10\cos(0)) \right] = (-10(-1))-10(1) = 20-10 = 10 $$

Step 4: Calculate the average height

Now, we need to divide this integral value by the length of the interval to find the average height: $$ \text{Average height} = \frac{\int_{0}^{\pi} 10 \sin x \, dx}{\pi - 0} $$ Plug in the integral value we found in Step 3: $$ \text{Average height} = \frac{10}{\pi} $$

Step 5: Interpret the result

The average height of the arch above the ground is \(\frac{10}{\pi}\) units.

Key Concepts

Definite Integrals

Definite integrals are a foundational concept in calculus, representing the net area under a curve between two points on the x-axis. The area can represent various physical quantities depending on the context, such as distance traveled, volume, or, as in our example, the total height across the span of an arch. To find this, we integrate the function over a specific interval, say from a to b, using the notation \[\int_{a}^{b} f(x) \, dx\].

When applying definite integrals to real-world problems like the average height of an arch, they offer a precise method to accumulate infinitely small pieces of height across the span to find a total. By evaluating the integral, we are essentially summing up all these tiny segments of area under the curve from the starting point a to the end point b. In our arch height problem, we calculated the definite integral of \(10 \sin x\) from \(0\) to \(\pi\), which summed up all of the heights from one end of the arch to the other.

Trigonometric Functions

In mathematics, trigonometric functions like sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)) are vital tools to model periodic phenomena such as sound waves, light waves, and, in our case, the shape of an arch. These functions relate the angles of a triangle to the lengths of its sides in a right-angled triangle but also define wave-like patterns when graphed as functions.

For the arch height problem, the sine function \(\sin x\) shapes the curvature of the arch—rising and falling smoothly between its maximum and minimum values. The function \(y = 10 \sin x\) gives us a wave that peaks at \(10\) and ebbs to \(-10\), depicting the height at various points along the arch from \(0\) to \(\pi\), scaled by a factor of \(10\) to represent the arch’s actual height. Grasping how trigonometric functions graph out and how their integrals work is essential for solving many practical problems in physics and engineering.

Integral Calculus

Integral calculus is the branch of mathematics that deals with integrals and their properties. It's used to find areas, volumes, central points, and many useful things, but it's not just about geometric shapes. Integral calculus joins (integrates) small pieces together to find how much there is of something, like adding up the slices to find the whole.

In our examination of the arch's height, integral calculus comes into play by integrating the height function over a certain interval. The operation of integration takes the basic premise of summing small parts to calculate quantities that are not easily measured by traditional methods. The process consists of finding the antiderivative of the function and evaluating this at the endpoints of the interval, which is then applied to various principles such as calculating the average value of a function over a specific domain—exactly the method used to determine the average height of the arch.

Average Value of a Function

The average value of a function over a given interval is a central theme in many areas of applied mathematics. It is defined as the integral of the function divided by the length of the interval and answers the question: 'What is the single value that represents the function’s output on average over the interval?'

In relation to the arch height problem, the average height is like asking