Problem 46 - Calculus: One and Several Variables

Question

The region between the curve \(y=\cos x\) and the \(x\) -axis from \(x=-x / 2\) to \(x=\pi / 2\) is revolved about the \(x\) -axis. Find the volume of the resulting solid.

Step-by-Step Solution

Answer

The volume of the solid generated by revolving the region between the curve \(y = \cos x\) and the x-axis from \(x = -\frac{x}{2}\) to \(x = \frac{\pi}{2}\) around the x-axis is given by the expression: \[V = \pi \left[ \frac{1}{2} (\frac{\pi+x}{2}) -\frac{1}{4}\sin x \right]\]

Step 1: Define the equation for the area of the circular disk

The equation we need to integrate is the area of the circular disk. Since the radius of the disk is represented by the curve, in this case, \(y = \cos x\), the area of the circular disk is: \[A(x) = \pi [(\cos x)^2]\]

Step 2: Set up the integral for volume

The volume of the solid is found by integrating the area function along the x-axis from -\(\frac{x}{2}\) to \(\frac{\pi}{2}\). This can be written as: \[V = \int_{-\frac{x}{2}}^{\frac{\pi}{2}} A(x) dx\]

Step 3: Substitute the area function into the integral

Replace the area function, \(A(x)\), with its equation from step 1: \[V = \int_{-\frac{x}{2}}^{\frac{\pi}{2}} \pi [\cos^2 x] dx\]

Step 4: Solve the integral

To solve the integral, use the power reduction formula for \(\cos^2x\): \[\cos^2 x = \frac{1 + \cos(2x)}{2}\] Now replace \(\cos^2x\) in the integral with the power reduction formula: \[V = \int_{-\frac{x}{2}}^{\frac{\pi}{2}} \pi \left[ \frac{1 + \cos(2x)}{2} \right] dx\] Now expand the integral and integrate with respect to x: \[V = \pi \int_{-\frac{x}{2}}^{\frac{\pi}{2}} \left[ \frac{1 + \cos(2x)}{2} \right] dx = \pi \left[ \int_{-\frac{x}{2}}^{\frac{\pi}{2}} \frac{1}{2} dx + \int_{-\frac{x}{2}}^{\frac{\pi}{2}} \frac{\cos(2x)}{2} dx \right]\] Now integrate both parts: \[V = \pi \left[ \frac{1}{2} \int_{-\frac{x}{2}}^{\frac{\pi}{2}} 1 dx + \frac{1}{2} \int_{-\frac{x}{2}}^{\frac{\pi}{2}} \cos(2x) dx \right]\] \[V = \pi \left[ \frac{1}{2} (x) \bigg|_{-\frac{x}{2}}^{\frac{\pi}{2}} + \frac{1}{2} \frac{1}{2} \sin(2x) \bigg|_{-\frac{x}{2}}^{\frac{\pi}{2}} \right]\]

Step 5: Evaluate the integral

Evaluate the definite integral: \[V = \pi \left[ \frac{1}{2} (\frac{\pi}{2} - (-\frac{x}{2})) + \frac{1}{2} \frac{1}{2} (\sin(\pi) - \sin(-x)) \right]\] Since \(\sin(\pi) = 0\) and \(\sin(-x) = -\sin x\): \[V = \pi \left[ \frac{1}{2} (\frac{\pi+x}{2}) -\frac{1}{4}\sin x \right]\] This expression represents the volume of the solid generated by revolving the region between the curve \(y = \cos x\) and the x-axis from \(x = -\frac{x}{2}\) to \(x = \frac{\pi}{2}\) around the x-axis.

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of mathematics involving the process of integration, an operation that amalgamates small data pieces into a whole. As the inverse operation to differentiation, it computes the accumulation of quantities, such as areas under a curve, total distance traveled, or the mass of an object with varying density.

When solving problems in physics and engineering, we often use integral calculus to find values associated with changes. For instance, in determining the work done by a force, we integrate the force function with respect to the distance.

Volume by Integration

In the realm of integral calculus, calculating the volume of a solid of revolution is a classic application. A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional curve around an axis. The volume by integration method involves slicing the solid into infinitesimally thin disks or washers, calculating the area of a slice, and then integrating this area over the interval of interest.

Example of Volume Calculation

For the exercise given, the method of integration computes the volume of the solid formed by revolving the region defined by the curve y = cos x. By integrating the area of circular slices perpendicular to the x-axis, the total volume is obtained.

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, tied intrinsically to the study of right-angled triangles and describing phenomena like sound, light waves, and oscillatory motions. The six main trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

In the context of the given exercise, the cosine function, represented as y = cos x, determines the radius of each circular disk being used to calculate the volume of the solid of revolution. The cosine function oscillates between -1 and 1, and is crucial in modelling periodic behaviour.

Power Reduction Formula

Power reduction formulas are a set of trigonometric identities that simplify powers of sine and cosine to the first power. These formulas are invaluable for integrating trigonometric functions, which appear frequently in calculus. Specifically, the formula \[\cos^2 x = \frac{1 + \cos(2x)}{2}\] simplifies the integral of \cos^2 x into a more manageable form, making the integration process straightforward.

In our exercise, the power reduction formula is utilized to integrate cosine squared, transforming the integral into a sum of simpler integrals. This process highlights how trigonometric identities can ease the computation of volumes of solids of revolution.