Question

In Exercises \(57-60\), use the Theorem of Pappus to find the volume of the solid of revolution. The solid formed by revolving the region bounded by the graphs of \(y=x, y=4,\) and \(x=0\) about the \(x\) -axis

Short answer

The volume of the formed solid is \(\frac{64}{3}\pi\) cubic units.

Step-by-step solution

Determine the Bounded Region

The given equations form a bounded region which is a right triangle with vertices at (0,0), (0,4), and (4,4). The base of this triangle is along the y-axis and ranges from y=0 to y=4, with the height being the line \(y=x\) from x=0 to x=4. We will revolve this region about the x -axis.

Calculate the Area of the Region

As described, the region is a triangle. The area \(A\) of a triangle is given by the formula \(\frac{1}{2} * base* height\). Here, base = height = 4. Hence, \(A = \frac{1}{2} * 4 * 4 = 8\) square units.

Identify the Centroid of the Region

The centroid of a triangle is located at \(\frac{2}{3}\) of the distance from the vertex to the midpoint of the opposite side. Since the triangle in this problem is a right triangle with the vertex at the origin, the centroid is \(\left(\frac{2}{3}*\frac{height}{2},\frac{2}{3}*\frac{base}{2}\right)\) which simplifies to \(\left(\frac{4}{3},\frac{4}{3}\right)\). Thus, the distance between the x-axis and the centroid is \(y = \frac{4}{3}\) units.

Apply Pappus’s Theorem

Apply Pappus’s Theorem to find the volume \(V\). The theorem states that \(V = 2πR*A\), where \(R\) is the distance from the centroid to the axis of rotation. So, \(V = 2* \pi * \frac{4}{3} * 8\). After simplification, \(V = \frac{64}{3}π\) cubic units.

Key concepts

Solid of Revolution

A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional shape around an axis. The resulting solid has a circular symmetry, and its volume can be found using various methods, one of which involves the Theorem of Pappus. This theorem is particularly useful as it reduces the problem of volume calculation to determining the area of the shape being revolved and the path of its centroid once rotated.

For instance, when we rotate a right triangle about an axis, we are creating a cone-shaped solid. To solve this using Pappus's Theorem, we need to find the area of the original triangle and the path the centroid takes during rotation, which is a circle. The volume is then simply the product of the area, the path of the centroid (a circle), and a constant, which connects to the formula volume equals the cross-sectional area multiplied by the centroid's path length.

Centroid of a Region

The centroid is the 'center of mass' of a shape, or in geometrical terms, it is the average position of all the points in the area of the shape. In our triangle example, the centroid is the point where the medians (the lines drawn from each vertex to the midpoint of the opposite side) intersect.

To determine the centroid of a simple shape like a triangle, there is a straightforward formula. However, for more complex shapes, we might need to use integral calculus to find it. The centroid plays a critical role in the application of Pappus's Theorem; it's the 'R' in the volume formula, representing the distance from the rotational axis to the centroid.

Area of a Triangle

The area of a triangle is one of the most fundamental concepts in geometry. It's given by the formula \( A = \frac{1}{2} * base * height \). In our problem, the triangle is right-angled and has equal base and height, which simplifies the calculation significantly.

Knowing how to compute the area is essential not only for geometry problems but also when dealing with solids of revolution since the cross-sectional area directly influences the volume. For triangles with different angles or side lengths, there are other formulas, such as Heron's formula, which might be used depending on the given information.