Problem 39 - Calculus Early Transcendentals

Question

Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by \(y=x^{2}, y=4,\) and \(x=0\) is revolved about the following lines. $$y=6$$

Step-by-Step Solution

Answer

The volume of the solid is \(\frac{36}{5}\pi\) cubic units.

Step 1: Identify the Outer and Inner Radii

To use the washer method, we need to first find the outer and inner radii at each point on the curve. The outer radius is the distance from the line of reflection, \(y=6\), to the top of the region we want to rotate, \(y=1\). So the outer radius R is 5. The inner radius is the distance from the line of reflection, \(y=6\), to the bottom of the region we want to rotate, \(y=x^2\). Therefore, the inner radius r is \(6 - x^2\).

Step 2: Set up the Washer Method Integral

To find the volume of the solid using the washer method, we need to integrate over the interval that the region is bounded by, which is \(x=0\) to \(x=1\). The volume of the solid can be found by subtracting the inner radius squared from the outer radius squared, multiplied by the height and \(\pi\). In our case, the height is constant, and is the distance between \(y=6\) and \(y=1\), which is \((6-1)=5\). The washer method integral is: $$V = \pi \int_{0}^{1} [(5^2) - ((6-x^2)^2)] dx$$

Step 3: Simplify and Calculate the Integral

Now, we need to simplify the integral and calculate the volume: $$V = \pi \int_{0}^{1} [25 - (36 - 12x^2 + x^4)] dx$$ $$V = \pi \int_{0}^{1} [-11 + 12x^2 - x^4] dx$$ Now, we'll integrate with respect to x: $$V = \pi \left[-11x + 4x^3 - \frac{1}{5}x^5\right]_{0}^{1}$$ Next, we'll evaluate the antiderivative at the bounds of integration: $$V = \pi \left\{-11(1) + 4(1)^3 - \frac{1}{5}(1)^5\right\} - \pi \left\{-11(0) + 4(0)^3 - \frac{1}{5}(0)^5\right\}$$ $$V = \pi \{-11 + 4 - \frac{1}{5}\}$$ $$V = \pi \{-7 - \frac{1}{5}\}$$ $$V = -\pi\left( 7 + \frac{1}{5} \right)$$ $$V = -\frac{36}{5}\pi$$ Since we want the volume, we need the absolute value of this result: $$V = \frac{36}{5}\pi$$

Step 4: Final Answer

The volume of the solid generated when the region in the first quadrant bounded by \(y=x^2, y=1,\) and \(x=0\) is revolved about the line \(y=6\) is \(\frac{36}{5}\pi\) cubic units.

Key Concepts

Washer Method

The Washer Method is a fascinating tool in calculus used to find the volume of solids generated by rotating a region around a line.
This technique is particularly useful when the solid has a hole, much like how a washer has a hole in the center.
In the washer method, you calculate the volume by considering the outer radius and inner radius of the solid.

• **Outer radius**: The distance from the axis of rotation to the outer curve of the solid.
• **Inner radius**: The distance from the axis of rotation to the inner curve or hole.

With these in hand, the volume is calculated using the integral:\[V = \pi \int_{a}^{b} \left(R^2 - r^2\right) \, dx\]Here, \(R\) and \(r\) are the outer and inner radii, respectively. This formula captures the idea of finding the volume of the larger outer shape and subtracting the unwanted volume of the inner shape.
This allows for accurate calculations of real-life objects that closely resemble washers.

Shell Method

The Shell Method provides an alternative way to calculate the volume of solids of revolution.
It's especially convenient when rotating a shape around the axis not defined along the variable of integration.
Unlike the washer method, which considers disks and holes, the shell method uses cylindrical shells to find volume.
The basic idea is to break down the solid into cylindrical shells and sum their volumes together. This is particularly effective when dealing with functions where establishing a distinct innermost and outermost point is easier than distinguishing inner and outer radii.
The integral for the shell method is formulated as:\[V = 2\pi \int_{a}^{b} x(f(x)) \, dx\]

• **Height of the shell**: Represented by \((f(x))\), it's the distance between the top and bottom functions at a given point.
• **Radius of the shell**: Given by \(x\), the distance to the axis of rotation.

This method is particularly advantageous for functions or shapes that extend vertically and helps simplify calculations when the axis of rotation isn't parallel to the primary faces of the shape.

Integration

Integration is a core concept in calculus essential to calculating the area under curves, the slope of a tangent line, and also volumes of solids of revolution.
The process of integration involves finding anti-derivatives, which are functions describing the total growth accrued by a function over an interval.
In the context of volumes of revolution:

• **Definite integrals** are used to evaluate the total accumulated volume across a specified interval.
• **Limits of integration** define these boundaries through which the integration is performed.

Using integration, complex shapes and regions can be broken down into infinitesimally small pieces where sums of these parts give us comprehensive results, like total area or volume.
This important branch of calculus provides tools such as the Fundamental Theorem of Calculus, which connects differentiation and integration, offering a strong foundation for analyzing and understanding real-world problems.

Calculus

Calculus is the branch of mathematics that deals with change.
It includes differential calculus, which focuses on rates of change, and integral calculus, which looks at accumulation of quantities.
A distinctive feature of calculus includes limits, which allow us to define concepts such as derivatives and integrals precisely.
This has enabled calculus to become crucial for solving problems across physics, engineering, economics, and beyond.

• **Differential Calculus**: Deals with instantaneous rates of change and slopes of curves.
• **Integral Calculus**: Concerned with the accumulation of quantities and the areas under and between curves.

In tasks like finding volumes with the washer or shell method, calculus leverages integrals to solve for values that describe how shapes interact, expand, and rotate in space.
This not only aids in answering theoretical questions in mathematics but also applies to real-world scenarios, manifesting in anything from calculating fluid flow to modeling planetary orbits.