Question

For the region bounded by the graphs of the equations, find: (a) the volume of the solid formed by revolving the region about the \(x\) -axis and (b) the centroid of the region. $$ y=\cos x, y=0, x=0, x=\pi / 2 $$

Short answer

The volume of the solid formed by revolving the region about the \(x\) -axis is computed by evaluating the integral \(\pi \int_{0}^{\pi/2} \cos^{2}x dx\). The centroid of the region is computed by evaluating \(\frac{1}{A} \int_{0}^{\pi/2} x \cos x dx\) for \(\bar{x}\) and \(\frac{1}{2A} \int_{0}^{\pi/2} \cos^{2}x dx\) for \(\bar{y}\), where \(A = \int_{0}^{\pi/2} \cos x dx\).

Step-by-step solution

Plot the Bounded Region

Using a graphing utility, plot \(y = \cosx\), \(y=0\), \(x=0\), and \(x= \pi /2\). The region in question is the area bounded by these four equations.

Determining Volume of Solid

After getting a visual representation of the region, we now compute the volume of the solid formed by revolving the region about the x-axis. The formula to calculate this is: \[V = \pi \int_{a}^{b} [f(x)]^2 dx\] where the bounds \(a\) and \(b\) are \(0\) and \(\pi / 2\) respectively and \(f(x) = \cos x.\] Now, calculate the integral \(\pi \int_{0}^{\pi/2} cos^2 x dx\).

Compute Centroid of the Region

The centroid of the region bounded by the aforementioned equations can be calculated using the centroid formula \((\bar{x}, \bar{y})\), where \(\bar{x} = \frac{1}{A} \int_{a}^{b} x f(x) dx\), \(\bar{y} = \frac{1}{2A} \int_{a}^{b} [(f(x))^2 dx]\) and \(A = \int_{a}^{b} f(x) dx\). Substituting \(f(x) = \cos x\) and bounds \(a = 0\) and \(b = \pi / 2\) into these formulas, perform the calculations to find out the coordinates of the centroid.

Key concepts

Volume of Solid of Revolution

When a two-dimensional region is rotated about an axis, the resulting three-dimensional object is known as a solid of revolution. Calculating the volume of these solids is a fundamental application of integral calculus. Imagine a shape being spun around an axis; the volume of the solid formed can be found using the disk or washer method.

In our exercise, the solid's volume is determined by revolving the area bounded by the equations around the x-axis. The region is defined by the curves of the cosine function, the x-axis, and the vertical lines at 0 and \(\pi/2\). To compute this, the disk method involves integrating the area of circular slices perpendicular to the axis of rotation. The formula for a solid rotated around the x-axis is \[V = \pi \int_{a}^{b} [f(x)]^2 dx\], where \(f(x)\) is the function that defines the upper boundary of the region being revolved. This formula uses the definite integral to accumulate the volume of infinitesimally thin disks from \(a\) to \(b\), the interval over which the region extends.

Centroid of a Region

The centroid, often referred to as the geometric center, is a point that represents the average position of all the points within a shape. For planar regions, the centroid coordinates \(\bar{x}\) and \(\bar{y}\) can be found using specific integrals that consider the shape's geometry and density.

The exercise in question involves determining the centroid of the region enclosed by the cosine function and the x-axis between 0 and \(\pi/2\). In calculus, the centroid of a uniform density region can be calculated using the formulas: \[\bar{x} = \frac{1}{A} \int_{a}^{b} x f(x) dx\] and \[\bar{y} = \frac{1}{2A} \int_{a}^{b} [f(x)]^2 dx\], where \(A\) is the area under the curve \(f(x)\) from \(a\) to \(b\). These integrals essentially find the 'balance point' of the region by considering the distances of the region's points from the axis of symmetry.

Definite Integrals

Definite integrals are the backbone of solving many calculus problems involving area, volume, and centroid calculations. A definite integral represents the signed area under a curve bounded by two specified limits. It is depicted as \[\int_{a}^{b} f(x) dx\], where \(f(x)\) is a function and \(a\) and \(b\) are the lower and upper limits of integration, respectively.

In calculus, the definite integral is evaluated using the fundamental theorem, which links antiderivatives to area calculations. In the context of our centroid and volume calculations, the definite integral quantifies areas and accumulates quantities like 'slices' of volume or moments for centroid calculations along an interval from \(a\) to \(b\). Understanding how to set up and evaluate definite integrals is crucial to solving these types of problems in calculus.

Graphs of Equations

Graphs are visual representations of equations and play a crucial role in understanding calculus problems by providing a geometric perspective of functions and the regions they define. Plotting graphs serves as a preliminary step that aids in visualizing the area under a curve, intersections, and the behavior of functions within specific intervals.

In the exercise at hand, graphing the functions involved, namely \(y = \cos x\), the x-axis \(y=0\), and the vertical lines \(x=0\) and \(x=\pi / 2\), demarcates the region related to our volume and centroid computations. Graphically examining these equations allows us to better understand the shape and limits of the region, ensuring accuracy as we proceed to calculate areas, volumes, and centroids using integration techniques.