Problem 69 - Calculus of a Single Variable

Question

Think About It Match each integral with the solid whose volume it represents, and give the dimensions of each solid. (a) Right circular cylinder (b) Ellipsoid (c) Sphere (d) Right circular cone (e) Torus $$ \begin{array}{l}{\text { (i) } \pi \int_{0}^{h}\left(\frac{r x}{h}\right)^{2} d x} \\ {\text { (iii) } \pi \int_{-r}^{r}\left(\sqrt{r^{2}-x^{2}}\right)^{2} d x} \\ {\text { (iv) } \pi \int_{-b}^{b}\left(a \sqrt{1-\frac{x^{2}}{b^{2}}}\right)^{2} d x} \\ {\text { (v) } \pi \int_{-r}^{r}\left[\left(R+\sqrt{r^{2}-x^{2}}\right)^{2}-\left(R-\sqrt{r^{2}-x^{2}}\right)^{2}\right] d x}\end{array} $$

Step-by-Step Solution

Answer

The matches are: \n(i)-Cone; radius \(r\), height \(h\) \n(iii)-Sphere; diameter \(2r\) \n(iv)-Ellipsoid; lengths of axes in the \(y\) and \(x\) directions as \(2a\) and \(2b\) respectively \n(v)-Torus; diameter of the torus \(2R\) and diameter of the tube \(2r\)

Step 1: Understand the formulas of volume for different shapes

Recall that the volume for different shapes in 3D are calculated in different ways: Cylinder \((\pi r^2 h)\), Ellipsoid \((\frac{4}{3}\pi abc)\) where \(a, b, c\) are the semi-axes of the ellipsoid perpendicular to the coordinate axes, Sphere \((\frac{4}{3}\pi r^3)\), Cone \((\frac{1}{3}\pi r^2 h)\) and Torus \((2\pi^2 Rr^2)\) where \( R \) is the distance from the center of the tube to the center of the torus and \( r \) is the radius of the tube.

Step 2: Interpret the integrals and match with corresponding shapes

(i) The integral represents the volume of a Cone, obtained by integrating the area of circles of continuously varying radii, which is a characteristic of a Cone. With dimensions: \(r\) as radius and \(h\) as height. (iii) This integral represents the volume of a Sphere. It's obtained by integrating the area of circles of continuously varying radii, but unlike in the Cone, the radii increase and then decrease symmetrically around \(x = 0\), which is a characteristic of a Sphere. With dimensions: \(2r\) as diameter. (iv) It represents the volume of an Ellipsoid. It's obtained by integrating the area of ellipse of continuously varying semi-axes, which is a characteristic of an Ellipsoid. With dimensions: \(2a\) as the length of axis in \(y\) direction and \(2b\) as the length of axis in \(x\) direction. (v) This integral represents the volume of a Torus. The expression being integrated is the difference of the squares of outer and inner radii of cross-sections of the torus, which varies symmetrically around \(x = 0\). This phenomenon is characteristic to a Torus. With dimensions: \(2R\) as diameter of the torus and \(2r\) as diameter of the tube.

Key Concepts

Integral Calculus

Integral calculus plays a pivotal role in calculating the volumes of various solids by allowing us to sum infinitely small quantities to find the total value. Think of it as a way to add up slices of a solid to form the complete shape. When we calculate the volume using integrals, we usually revolve a known shape or function around an axis and integrate over the specified range to find the solid's volume.

For instance, to find the volume of a cone or sphere, we can integrate the area of circles with varying radii, as seen in the textbook example. The integral takes all these individual areas and sums them up, giving us the volume of the respective 3D shape. It's the continuous nature of these slices that allows us to use integration to calculate the volume of objects that may have complex or changing shapes.

Geometric Solids

Geometric solids are three-dimensional objects with width, depth, and height, such as spheres, cones, and cylinders. Each of these shapes has distinguishing features based on their cross-sections and surfaces. A solid, such as a cone, will have a circular base and taper off to a point, while a sphere is perfectly symmetrical in all directions from its center.

In geometric terms, these solids are defined by their vertices, edges, faces, and the way these parts are connected or curved. The properties of these geometric solids help determine the approach we take with integral calculus when calculating their volumes. For example, a sphere's symmetry allows us to use a particular integration method— revolving an area around an axis—to simplify the calculation process and obtain the volume.

Volume Formulas

Each geometric solid has a corresponding volume formula that allows us to calculate its space. These formulas are derived based on the shape's specific properties and dimensions. A cylinder's volume, for instance, is the product of its base area (a circle) and its height, represented by the formula \(\pi r^2 h\).

For more complex shapes like an ellipsoid or a torus, the volume formulas become more intricate, as seen with the ellipsoid \(\frac{4}{3}\pi abc\) where \(a, b, c\) are the elliptical radii, and the torus \(2\pi^2 Rr^2\), which factors in both the radius of the tube (\(r\)) and the distance from the tube's center to the torus's center (\(R\)). Understanding these formulas is essential when studying calculus, as they often form the base expressions that we integrate to find the volumes in more advanced problems.