Problem 49 - Essential Calculus

Question

The integral represents the volume of a solid of revolution. Identify (a) the plane region that is revolved and (b) the axis of revolution. $$ 2 \pi \int_{0}^{6}(y+2) \sqrt{6-y} d y $$

Step-by-Step Solution

Answer

(a) The plane region is the area under the curve \(y = (y+2) \sqrt{6-y}\) from 0 to 6. (b) The axis of revolution is the y-axis.

Step 1: Identification of Function within the Integral

Look at the integrand within the integral \((y+2) \sqrt{6-y}\). This is the function that's being revolved around an axis to form a solid. It suggests that the curve described by \(y = (y+2) \sqrt{6-y}\) is being revolved.

Step 2: Recognize Limits of the Integral

The limits of the integral are from 0 to 6. Those represent the region in the x-axis from 0 to 6.

Step 3: Identify the Axis of Revolution

In the formula for solid revolution, 2π times integral, when we have y in terms of only x or a constant the solid is revolved around the x-axis. Here since the function is in terms of 'y', it is revolved around the y-axis.

Key Concepts

Volume of Solid of Revolution

Understanding how to calculate the volume of a solid of revolution is an essential skill in calculus. A solid of revolution is created when a region in the plane is revolved around a line, which we refer to as the axis of revolution. The integral provided in the exercise, \( 2 \pi \int_{0}^{6}(y+2) \sqrt{6-y} dy \), is a mathematical representation of this concept.

The formula incorporates the definite integral to calculate the volume. The function \( (y+2) \sqrt{6-y} \) describes the shape of the region that's being rotated. By integrating this function, we're essentially summing up an infinite number of infinitesimally thin discs or washers to find the total volume of the solid. The constant \(2\pi\) is a part of the formula for the circumference of a circle, which is crucial since each cross section of the solid is circular due to the rotation.

To fully understand this concept, picture a clay slab with the shape described by the function. If you were to spin this slab around a pole (the axis of revolution), the form it takes is the solid of revolution we're calculating the volume for. This practical image can help solidify the student's understanding of the abstract integral calculation.

Definite Integral

The definite integral is like a highly efficient accountant who precisely calculates the total of continuous transactions over a specific period. In mathematical terms, it sums up an infinite number of infinitesimal quantities over an interval, which can represent areas, volumes, and other physical concepts.

In the context of our exercise, the definite integral \( \int_{0}^{6}(y+2) \sqrt{6-y} dy \) calculates the volume as we've discussed. The limits of integration, 0 and 6, denote the boundaries of the region over which we are integrating. The lower limit, 0, indicates the start of the interval on the y-axis, and 6 represents the end. The computation within this range provides the exact volume of the solid of revolution.

Interpretation of Results

A key point to note is that the definite integral gives us a concrete number representing a physical quantity. It moves beyond theoretical expressions and offers us a means to calculate exact values which can be very empowering for a student grappling with the abstract nature of calculus.

Axis of Revolution

Imagine the axis of revolution as the spine of a book, with the pages representing cross-sectional slices of our three-dimensional solid. In the given exercise, we revolve the plane region around the y-axis. This axis is the line about which the two-dimensional region spins like a merry-go-round to create a three-dimensional object.

The clue within the integral \(2\pi\) suggests that we are dealing with circular motion, akin to wrapping the plane region around a cylinder. Differentiating between revolving around the x-axis and the y-axis is crucial in setting up the integral. When the function within the integral is expressed in terms of 'y', it indicates the solid is generated by revolving around the y-axis, which is the case in our exercise.

Choosing the Correct Axis

Understanding the axis of revolution also aids in visualizing the solid which can be quite challenging. It determines the shape and dimensions of the cross-sections used in forming the solid. This visual understanding is vital for comprehending the geometry and assembling the pieces of the puzzle that is solids of revolution in calculus.