Question

In Exercises 21 and 22, find the volume of the solid generated by revolving the region about the given line. the region in the first quadrant bounded above by the line \(y=2\) below by the curve \(y=2 \sin x, 0 \leq x \leq \pi / 2,\) and on the left by the \(y\) -axis, about the line \(y=2\)

Short answer

The volume of the solid generated is given by evaluating the integral: \(V = \pi \int_0^{\pi/2} (4 - 8\sin x + 2 - 2cos2x)\,dx\), which after integration and evaluation, provides the volume.

Step-by-step solution

Definition of Solid and Bounded Regions

The solid is defined by the region in the first quadrant bounded above by the line \(y=2\), below by the curve \(y=2 \sin x, 0 \leq x \leq \pi / 2\), and on the left by the y-axis. This region will be rotated about the line \(y=2\).

Disk Method Setup

The disc method formula involves integrating over the interval, the area of each disc formed. In general, this formula is \(\int_a^b π (f(x))^2\,dx\), where \(f(x)\) is the radius of the disc. In this case, the radius of each disc is the distance from the curve \(y=2\sin x\) to \(y=2\). That is \(2 - 2 \sin x\). The interval of rotation is from \(x = 0\) to \(x = \pi / 2\).

Using Disk Method to Solve for the Volume

Substituting into our volume formula, we get that the volume is: \(V = \int_0^{\pi/2} \pi (2 - 2\sin x) ^2\,dx\).

Evaluate the Integral

Evaluating this integral will give us the volume of the solid. The volume is \(V = \pi \int_0^{\pi/2} (4 - 8\sin x + 4sin^2x)\,dx\). Changing \( 4sin^2x \) to \( 4(1-cos2x)/2 \) using double angle formula, now we have \(V = \pi \int_0^{\pi/2} (4 - 8\sin x + 2 - 2cos2x)\,dx\). Integrating and evaluating at the limits should provide the volume of the solid.

Key concepts

Disk Method

To comprehend the disk method, envision slicing a solid of revolution into thin disks perpendicular to the axis of revolution. Imagine the solid standing still while a sharp blade cuts it into thin, round pieces, resembling disks.

Each disk's thickness is an infinitely small segment of the axis, denoted by a differential, say, \(dx\) if the axis is horizontal, or \(dy\) if it's vertical. The disk method then involves calculating the volume of each disk by using the area of a circle, \(\pi r^2\), where \(r\) is the radius at a specific point, determined by a function \(f(x)\) or \(g(y)\).

The total volume of the solid is found by integrating the volume of these disks over the given interval. In essence, the disk method sums an infinite number of infinitesimally thin disks to find the solid's total volume. It's perfect for solids with symmetrical cross-sections around the axis.

Solid of Revolution

A solid of revolution is a 3-dimensional object obtained by rotating a 2-dimensional shape around an axis. The shape you rotate could be a simple curve, a line, or an area bounded by several functions on a coordinate system.

For example, if you take a semicircle and rotate it around its diameter, you'll get a sphere. Similarly, rotating a rectangle around one of its sides gives you a cylinder. The shape after revolution has an axis of symmetry, which is the line or axis around which the rotation happens.

It's essential to vividly visualize the original 2-dimensional region before it transforms into the final solid. Your imagination needs to see the rotation and the resulting 3-dimensional figure, as this will aid in setting up the integral needed to calculate the volume of the solid correctly.

Definite Integral

The definite integral is an essential concept in calculus, representing the area under a curve over a specified interval. It is denoted as \(\int_a^b f(x)\text{ }dx\), where \(a\) and \(b\) are the limits of integration, marking the starting and ending points on the x-axis.

We compute the definite integral by finding the antiderivative of the function and then applying the Fundamental Theorem of Calculus, which involves subtracting the value of the antiderivative at \(a\) from its value at \(b\).

In the context of the solid of revolution, the definite integral isn't just an area calculation—it's now the sum of volumes of the infinitely many thin disks (or shells, in another method). Understanding the definite integral is crucial because it's the tool you use to move from a formula for the disk's volume to the entire solid's volume.

Trigonometric Functions

Trigonometric functions like sine and cosine are vital in connecting the geometry of circles to the algebra of equations. They frequently appear in problems involving periodic phenomena, right triangle relationships, and circles.

In the context of calculus, trigonometric functions help us describe curves, which we might rotate around an axis to create solids of revolution. For example, the function \(y=2\sin x\) describes a waveform, and when this curve is revolved about the line \(y=2\), it forms a solid.

Manipulating these functions, using identities such as the double-angle formula, can simplify integration when finding the volume of solids of revolution. In our solution, we convert \(4\sin^2 x\) into \(2 - 2\cos 2x\) to make the integral more straightforward to evaluate. Understanding such identities and the properties of trigonometric functions can greatly aid in solving complex calculus problems.