Problem 43 - Single Variable Calculus

Question

Use the Theorem of Pappus to find the volume of the given solid. The solid obtained by revolving the region bounded by the graphs of \(y=4-x^{2}, y=4\), and \(x=2\) about the \(y\) -axis

Step-by-Step Solution

Answer

The volume of the solid obtained by revolving the given region around the y-axis is \(\frac{32}{3}\pi\) cubic units, which was calculated using the Theorem of Pappus. First, we found the area of the region by integrating the vertical distance between the boundaries, which was \(\frac{16}{3}\). Then, we found the centroid's x-coordinate \(\bar{x}\) to be 1. Finally, we applied the Theorem of Pappus using the formula \(V = (2 \pi \bar{x}) \cdot A\) to determine the volume as \(\frac{32}{3}\pi\).

Step 1: Sketch the Region

To begin with, draw the graphs of \(y=4-x^2\), \(y=4\), and \(x=2\) on a coordinate plane, and identify the region enclosed by these graphs. You'll notice that the region is a part of the parabolic curve and is to the left of the vertical line \(x=2\).

Step 2: Find the Area of the Region

The area of the region can be found by integrating the vertical distance between the parabolic curve and the horizontal line \(y=4\). The equation of the parabolic curve can be written as \(x = \sqrt{4-y}\): \[A = \int_{a}^{b} [\sqrt{4-y}-0] dy\] To find the bounds for the integration, observe that the horizontal line \(y=4\) meets the parabolic curve \(y=4-x^2\) at the point when \(4=4-x^2\), thus, \(x=0\), the region starts at \(y=4\), then it goes down to \(y=0\) (where the curve intersects \(x=2\)-the x-axis). So, we integrate from \(a=0\) to \(b=4\): \[A = \int_{0}^{4} [\sqrt{4-y}] dy\]

Step 3: Evaluate the Definite Integral

Now we need to evaluate the definite integral to find the area of the region. Integration by substitution can be used. Let \(u = 4-y\) and \(du = -dy\). When \(y = 4\), \(u = 0\), and when \(y = 0\), \(u = 4\). Our integral becomes: \[A = \int_{4}^{0} [\sqrt{u}](-du)\] Switch the limits of integration to get rid of the negative sign: \[A = \int_{0}^{4} u^{1/2} du\] Now, integrate with respect to u: \[A = \frac{2}{3} u^{3/2} \Big|_0^4\] Plug in the bounds: \[A = \frac{2}{3} (4)^{3/2} - \frac{2}{3} (0)^{3/2} = \frac{16}{3}\]

Step 4: Find the Centroid

To find the centroid of the region, we need to find the average x and y values. Since we are revolving around the y-axis, the centroid's x-coordinate is the important one. To find the x-coordinate (\(\bar{x}\)) of the centroid, we use the formula: \[\bar{x} = \frac{1}{A} \int_{0}^{4} x(y) dy\] \[= \frac{1}{\frac{16}{3}} \int_{0}^{4} \sqrt{4-y} dy\] Remember that we have already found this integral is equal to \(\frac{16}{3}\): \[\bar{x} = \frac{3}{16} (\frac{16}{3}) = 1\] Now, the x-coordinate of the centroid is \(\bar{x}=1\).

Step 5: Apply the Theorem of Pappus

Now that we have found the area of the region and the location of the centroid, we can use the Theorem of Pappus to find the volume of the solid obtained by revolving the region about the y-axis: \[V = (2 \pi \bar{x}) \cdot A\] Plug in the values for the centroid and area: \[V = (2 \pi (1)) \cdot \left(\frac{16}{3}\right)\] \[V = \frac{32}{3}\pi\] Therefore, the volume of the solid is \(\frac{32}{3}\pi\) cubic units.

Key Concepts

Solid of Revolution

A solid of revolution is a three-dimensional figure created by rotating a two-dimensional shape around an axis. The result is a symmetrical object with a circular cross-section, like the shape of a doughnut, cylinder, or cone. For example, if you rotate a semicircle around its diameter, you'll end up with a sphere.

To visualize this, imagine spinning a line, curve, or region around an axis; every point of the shape traces a circular path, forming the surface of the solid. This principle is fundamental in fields of geometry and calculus when calculating volumes of irregularly shaped objects.

Volume Calculation

Calculating the volume of a solid of revolution can often be performed using the method of slicing, where the volume is approximated by the sum of the volumes of many thin disks or washers. However, in calculus, an integral can be used to represent this sum and precisely calculate the total volume.

Method of Disks and Washers

When the solid's cross-section is uniform across the axis of rotation, the disk method calculates the volume of each thin disk separately and integrates across the interval. If there's a hole or gap, the washer method is employed, accounting for the inner radius to subtract the empty space.

Definite Integral

In calculus, a definite integral calculates the accumulated value of a function over a specific interval. This powerful tool financial analysis to physics for computations that span areas, volumes, sums, and more.

With reference to the area under a curve, the definite integral is defined by the difference between the values of the antiderivative evaluated at the boundary points. Mathematically, it is the limit of a sum of areas of rectangles (Riemann sums) as the width of the rectangles approaches zero, providing the exact area under the curve between two points.

Centroid of a Region

The centroid of a region is the geometric center of that area, analogous to the center of mass of a body. If you had a thin, rigid sheet in the precise shape of the region and tried to balance it on a pencil tip, the balancing point would be the centroid.

In a 2D plane, the centroid coordinates are found by calculating the average values of the position coordinates (x, y) across the entire area. When dealing with composite shapes, centroids of individual parts may be calculated and averaged, sometimes weighted by each part's area to get the overall centroid.