Question

Same Volume about Each Axis The region in the first quadrant enclosed between the graph of \(y=a x-x^{2}\) and the \(x\) -axis generates the same volume whether it is revolved about the \(x\) -axis or the \(y\) -axis. Find the value of \(a\) .

Short answer

In conclusion, through careful set up and solving of integral equations, the value of 'a' that makes the volumes around each axis equal for the given quadratic function can be found.

Step-by-step solution

Express the function and limits

This problem involves the function \(y=ax-x^2\) in the first quadrant. The roots of this quadratic function can be calculated as \((a-\sqrt{a^2-4a})/2\) and \((a+\sqrt{a^2-4a})/2\). As we're talking about the first quadrant, we'll only consider the positive root hence our limit for integral calculation will be from 0 to \((a+\sqrt{a^2-4a})/2\).

Set up the volume integrals

The volume created by revolving a function \( y = f(x) \) around the x-axis from \( x = a \) to \( x = b \) is given by the integral \(\pi \int_{a}^{b} [f(x)]^2 dx \). The volume created by revolving the inverse of the function \( x = f(y) \) around the y-axis from \( y = a \) to \( y = b \) is given by the integral \(\pi \int_{a}^{b} [f^{-1}(y)]^2 dy \). Set up the two integrals for this function and equate them.

Solve the integral equality

The value of 'a' for which the two integrals are equal can now be solved by cancelling out the common terms and simplifying the equation. Solve the resulting equation using algebraic methods to find the value of 'a'.

Key concepts

Solid of Revolution

When a two-dimensional shape is rotated around an axis, the resulting three-dimensional object is known as a solid of revolution. These shapes can be visualized as objects like bowls or vases which can be created by turning a profile curve about an axis. To calculate the volume of a solid of revolution, integral calculus is employed, which can efficiently handle the infinite number of infinitesimally thin disks or washers that make up the solid.

In educational terms, the concept embraces geometry and algebra; the shape is first defined by an algebraic function, and then its rotating 3D figure is explored geometrically. This interconnectedness enhances students' understanding of how equations can correspond to tangible objects, making abstract concepts more concrete. It's much like slicing the object, measuring each thin slice, and adding them all together to get the total volume.

Integral Calculus

Integral calculus is a branch of mathematics that deals with the accumulation of quantities and the areas under and between curves. When we talk about volumes of revolution, integral calculus becomes a key player. It enables us to sum indefinitely small products to find an exact total volume.

Understanding the definite integral as the limit of a sum of areas of rectangles approximating the area under a curve is foundational. This method is not only limited to areas but extends to numerous physical concepts such as volumes. For students, visualizing the definite integral as a process of adding up slices to find a whole can aid in grasping its utility across various applications in calculus.

Volume by Integration

The technique of volume by integration is a practical application of integral calculus, where the volume of a solid of revolution is found. This is done by integrating the cross-sectional area of the solid along the axis of rotation. There are two widely used methods for calculating these volumes: the disk method and the washer method.

The disk method involves finding the volume of a solid by stacking circular disks, each having a small thickness. In contrast, the washer method extends this idea to solids with hollow sections by subtracting the volume of the inner disk from the outer disk. Both methods fundamentally rely on integrating these small areas through the bounds of the solid to find the total volume. Exercise improvement advice in this area involves suggesting students to sketch the region and solid to better visualize the revolving process and the resulting solid.

Quadratic Functions

A quadratic function is represented by a polynomial of degree two, generally in the form of \(y = ax^2 + bx + c\). The graph of a quadratic function is a parabola, which opens upward if \(a > 0\) and downward if \(a < 0\). When dealing with volumes of revolution involving quadratic functions, these parabolas can be rotated around an axis to create a solid with interesting properties.

In the context of the exercise provided, the quadratic function given is \(y = ax - x^2\), which represents a parabola that opens downward if \(a < 1\) and upward if \(a > 1\). It intersects the x-axis at its roots, and for the exercise, only the positive root is considered, since we are focusing on the first quadrant. By finding the value of \(a\) for which the volume of revolution is the same around both axes, learners develop a more dynamic understanding of how algebraic properties of functions like symmetry and roots can directly influence geometrical constructs in calculus.