Question

In Exercises 31-36, find an equation for the surface of revolution generated by revolving the curve in the indicated coordinate plane about the given axis. $$ z=3 y \quad y z \text { -plane } \quad y \text { -axis } $$

Short answer

The equation of the surface of revolution generated by revolving the given curve about the \(y\)-axis is \(x^2+z^2=(3y)^2\).

Step-by-step solution

understanding given curve

The given curve is \(z=3y\) in the coordinate plane \(yz\). This is a straight line passing through the origin with a slope of 3.

Finding the radius of revolution

Next, understand that the distance from the curve to the axis of revolution will be the radius of the surface of revolution. When revolving around the \(y\)-axis, the distance to a point (x, y, z) on the surface from the y-axis is simple: it's just the x-coordinate. However, in the \(yz\)-plane, the x-coordinate is not provided. But we do know it's a surface of revolution about the \(y\)-axis, which means the x-coordinate is the distance from the y-axis. The height \(z\) simply remains the same.

Formulating the surface equation

Finally, the equation for a surface of revolution (cylinder) formed by rotating the curve \(z=f(y)\) about the \(y\)-axis is \(x^2+z^2=[f(y)]^2\). Here, since \(f(y)=3y\), the equation of the surface is \(x^2+z^2=(3y)^2\).

Key concepts

Calculus

Calculus is an area of mathematics that deals with changes and motion. It is divided into two main branches: differential calculus, which focuses on rates of change and slopes of curves, and integral calculus, which focuses on the accumulation of quantities and the area under and between curves.

In the context of surfaces of revolution, calculus allows us to find and describe the shape of a 3D object created by revolving a 2D curve around an axis. This involves using integral calculus to compute volumes and surface areas of these solids. Calculus also assists in understanding the rate of change at any point on the surface, which can be important for applications such as fluid dynamics and structural engineering.

Moreover, understanding derivatives and integrals is essential when dealing with the equations that represent these surfaces. For students looking to master the creation and calculation of surfaces of revolution, a solid grasp of calculus is indispensable.

Equation of a Surface

The equation of a surface represents a relationship between the three spatial coordinates, usually denoted as (x, y, z). These equations are essential for visualizing and working with shapes in three-dimensional space, especially in areas such as solid geometry and vector calculus.

In our exercise, the equation of a surface is derived from a curve that has been revolved around an axis. For example, the curve given by the equation z=3y is revolved around the y-axis to form a surface. The equation of this surface of revolution is not explicit in x, y, and z, which makes deriving it a bit challenging. However, by using the symmetrical property of solids of revolution (also known as cylindrical symmetry), we can relate the x and z coordinates through the function f(y), which describes the original curve.

Therefore, by finding the radial distance (in this case, the x-coordinate) in terms of y and z, we can construct the equation for the entire surface. It's vital to comprehend these equations, as they serve as the basis for further calculations like volume, surface area, center of mass, and many other physical properties.

Solid of Revolution

A solid of revolution is a three-dimensional object obtained by revolving a two-dimensional curve around an axis. This process is akin to spinning the curve around a line (the axis) to create a shape with axial symmetry.

In our exercise, the curve z=3y, which lies in the yz-plane, is rotated around the y-axis. As this happens, each point on the curve traces out a circle in the xz-plane, and the ensemble of these circles forms the surface of the solid of revolution. The radius of each circle comes from the x-coordinate, which in this particular case, is the distance from each point to the axis of rotation. Therefore, using the formula for a surface of revolution, we can describe the solid mathematically with the equation x²+z²=(3y)².

This equation represents a cylinder whose cross-sections perpendicular to the y-axis are circles with a radius that changes in relation to the y-coordinate. By studying solids of revolution, students can better understand the geometric properties of shapes created by rotation, which has practical applications in engineering, architecture, and the physical sciences.