Problem 49 - Calculus of a Single Variable

Question

Describing a Solid In Exercises 49 and \(50,\) the integral represents the volume of a solid. Describe the solid. $$ \pi \int_{0}^{\pi / 2} \sin ^{2} x d x $$

Step-by-Step Solution

Answer

The solid represented by the integral \( \pi \int_{0}^{\pi / 2} \sin^{2} x dx \) is a symmetrical shape around the x-axis, most likely semi-cylindrical or semi-spherical. Its volume is \(\frac{\pi^{2}}{4}\).

Step 1: Identify the bounds and the function

Identify the bounds of the integral and the function that is being integrated. Here, the bounds are \(0\) to \(\pi/2\), and the function is \(\sin^{2}(x)\). The \(\pi\) outside the integral indicates we are dealing with a shape that has circular symmetry.

Step 2: Find the antiderivative

Before we attempt to visualize the solid, we need to find the integral. We see that the integral is of the form \( \sin^{2}(x) \) which can be written as \(\frac{1 - \cos(2x)}{2}\). Now the integral will be \(\pi \int_{0}^{\pi / 2} \frac{1 - \cos(2x)}{2} dx\). Using the power rule, the antiderivative becomes \(\frac{\pi}{2} [x - \frac{\sin(2x)}{4} ]_{0}^{\pi / 2}\).

Step 3: Evaluate the definite integral

Next, plug in the upper and lower limits of integration to evaluate the integral. This results in \(\frac{\pi}{2} * [\pi/2 - 0]\), which equals \(\frac{\pi^{2}}{4}\). This is the volume of the solid.

Step 4: step 4: Describe the solid

The given volume represents a solid that is symmetrical about the x-axis since the equation includes a π term. This symmetry, along with the presence of the sine function, indicates that the solid could be a semi-cylindrical or semi-spherical object, depending on the specifics.

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus. They help us determine the net area under a curve, bounded by two points on the x-axis. This is also how we calculate volumes of solids of revolution.

A definite integral is expressed with specific upper and lower limits, such as in \(\int_{a}^{b} f(x) dx\). These limits, \(a\) and \(b\), are the points between which we are interested in finding the area or volume. In the given exercise, the bounds are \(0\) to \(\pi/2\).

It's important to note that definite integrals not only help us find "area under the curve" but also aid in determining volumes when dealing with solids of revolution. This is a method where a region is rotated around an axis, forming a 3D shape, and the volume can be determined using integrals.

Antiderivatives

Antiderivatives, also known as indefinite integrals, are essentially the inverse operation of differentiation. When we find an antiderivative, we are essentially "going backwards" from the derivative to figure out what original function led to it.

In solving definite integrals, we often seek the antiderivative of the function inside the integral. For example, solving \(\int \sin^{2}(x) dx\), we rewrite sin-squared using a trigonometric identity to make it easier to integrate. This involves turning \(\sin^{2}(x)\) into \(\frac{1 - \cos(2x)}{2}\)\, simplifying the integration process.

After finding the antiderivative, we apply the limits to evaluate the definite integral. This specific step gives the real-world quantity we're seeking, like volume.

Trigonometric Functions

Trigonometric functions are among the most common types of functions encountered in calculus, particularly when calculating volumes and areas. Functions like sine, cosine, and tangent, model periodic behaviors, which can be extended to model circular and rotational motion in solids.

In the given problem, we deal with the function \(\sin^{2}(x)\). Sine squared appears often in physics and engineering problems where wave patterns or periodic motion around a point are modelled.

When we compute \(\int \sin^{2}(x) dx\), the integral involves a trigonometric identity: the half-angle identity. This transforms the problem into a simpler one using cosine. It shows how trigonometric identities are used effectively in calculus to solve integrals that initially appear complex.