Problem 30 - Calculus Early Transcendentals

Question

Let \(R\) be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=x, y=x+2, x=0, x=4$$

Step-by-Step Solution

Answer

Answer: The volume of the solid generated is 48π cubic units.

Step 1: Sketch the Region R

First, we need to sketch the region R. To do this, we plot the lines y=x, y=x+2, x=0, and x=4. These lines form a parallelogram, and it is this region that will be revolved about the x-axis to form our solid.

Step 2: Determine Inner and Outer Radii

Since the region is being revolved around the x-axis, we need to consider horizontal washers. The inner radius of the washer is the distance from the line y=x to the x-axis, and the outer radius of the washer is the distance from the line y=x+2 to the x-axis. Hence, $$Inner \: Radius = y_1 = x$$ $$Outer \: Radius = y_2 = x+2$$

Step 3: Set up the Integral

Now, we want to find the volume of the solid generated by integrating the area of the washers over the interval [0, 4]. For the washer method, the area of each washer is given by the difference of the square of the outer and inner radii multiplied by π. Therefore, we will set up our integral as follows: $$Volume = \pi\int_{0}^{4} [(x+2)^2 - x^2 ]dx$$

Step 4: Simplify the Integral

Now we simplify the integrand: $$Volume = \pi\int_{0}^{4} [(x^2+4x+4) - x^2 ]dx$$ $$Volume = \pi\int_{0}^{4} (4x + 4)dx$$

Step 5: Evaluate the Integral

Now we evaluate the integral by finding the antiderivative and applying the fundamental theorem of calculus. $$Volume = \pi \left [2x^2 + 4x \right]_{0}^{4} $$ $$Volume = \pi [(2(4)^2 + 4(4)) - (0)]$$ $$Volume = \pi[(32+16)]$$ $$Volume = 48\pi$$ Therefore, the volume of the solid generated when the region R is revolved about the x-axis is 48π cubic units.

Key Concepts

Volume

Volume is a critical concept in calculus and geometry. It measures the amount of space an object occupies in three dimensions. Understanding volume is essential when dealing with real-world applications, such as determining how much water a container can hold or how much material is needed to construct an object.
The volume of a solid of revolution, like the one in the given problem, is calculated by revolving a region around an axis. In this case, the volume was determined by revolving a region about the x-axis using a specific method (washer method) designed for finding volumes of such solids. Calculating volume using calculus involves setting up and solving an integral, which captures the accumulation of volume elements across the region of interest.

Washer Method

The washer method is a technique used in calculus to find the volume of a solid of revolution. When you have a region bounded by two curves that you wish to revolve around an axis, the washer method helps account for the space between the curves. This prevents simply mirroring the volume on the other side of the axis, instead accurately capturing the real volume in between.
The method involves visualizing the solid as a series of thin disks or washers stacked along the axis of rotation. Each 'washer' has an outer radius determined by the outer curve and an inner radius determined by the inner curve. By calculating the area of one such washer and then integrating over the range of interest, you find the entire volume. For example, in this problem:

• Inner radius: Distance from the x-axis to the lower boundary (
• Outer radius: Distance from the x-axis to the upper boundary

The next step involves setting up an integral that sums up the differences in squared radii across the given interval.

Revolution about an Axis

When learning about solids of revolution, the idea of revolving a shape (region) around a straight line, or axis, is fundamental. This process involves taking a 2D shape and spinning it around a line to create a 3D solid.
In the current solution, the region bounded by the curves was revolved around the x-axis. By doing this, a symmetrical 3D shape is created, similar in technique to pottery on a wheel. Understanding revolution about an axis is critical since it allows for the practical application in designing objects that require rotational symmetry, such as wheels, bowls, or bottles.
The choice of axis (x or y) greatly affects the solid's volume and geometry, highlighting the importance of accurately setting up your calculations and integral bounds to ensure correct results.

Integration

Integration is a core concept in calculus, acting as the counterpart to differentiation. While differentiation measures change, integration sums up quantities, capturing accumulation and area under curves. It's a powerful tool for finding quantities that cannot be easily measured through simple arithmetic.
In this washer method solution, integration was used to compute the volume of the revolved solid efficiently. By setting up an integral from 0 to 4 with respect to the x-axis, and incorporating the difference between the squares of the outer and inner radii, the total volume was obtained:
The integral:
\( \pi \int_{0}^{4} ((x+2)^2 - x^2) \, dx \)
represents the sum of all infinitesimally small volume elements, each captured by the area of a washer. The final solution involves evaluating this integral using the fundamental theorem of calculus, simplifying our integrand, and computing the result to find the exact volume of the shape.