Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\sqrt{x}, \quad y=-\frac{1}{2} x+4, \quad x=0, \quad x=8 $$

Short answer

The volume of the solid generated by revolving the bounded region between the curves \(y=\sqrt{x}\), \(y=-\frac{1}{2} x+4\), the line \(x=0\) and \(x=8\) around the x-axis can be calculated by setting up and evaluating a definite integral using the disk method.

Step-by-step solution

Find the intersection points

Set the two equations equal to each other, \(\sqrt{x}=-\frac{1}{2} x+4\) and solve for \(x\). Starting by squaring both sides to get rid of the square root: \(x = (\frac{-x}{2} + 4)^2\), which simplifies to the equation \(x = \frac{x^2}{4} - 4x + 16\). Solving this quadratic equation gives the intersection points at \(x=0\) and \(x=8\).

Visualize the area

Drawing the curves helps us understand the problem better and pick the right method for finding the volume of a solid of revolution. In this case, we can see that the area bounded by the curves and the x-values 0 and 8 is to be rotated about the x-axis, so we use the disk method to find the volume.

Apply Disk method

The Disk Method is given by the formula \[ V = \pi \int_{a}^{b} [R(x)]^2\, dx \] where \(R(x)\) is the radius of the disk at x. Here, \(R(x)\) is the height of the small rectangular slice which is equivalent to the difference between the two functions, \((-1/2)x + 4 - \sqrt{x}\). The limits of integration, \(a\) and \(b\), are from 0 to 8.

Evaluate the integral

Plug in the values into the integral: \[ V = \pi \int_{0}^{8} [(-1/2)x + 4 - \sqrt{x}]^2 dx \] Evaluating this definite integral results in the volume of the solid. Performing this computation leads to the final answer.

Key concepts

Disk Method

The Disk Method is a powerful technique for finding the volume of a solid of revolution, which is a solid created by rotating a two-dimensional region around an axis. This method involves slicing the solid perpendicular to the axis of rotation. Each slice is shaped like a disk (or a coin), hence the name.

Imagine cutting this solid into several thin disks, stacking them from the start of the region to the end. The volume of each tiny disk can be approximated by the volume of a cylinder, \( V_{disk} = \pi r^2h \) with h being the thickness of the disk that approaches zero (infinitesimally small) and r representing the radius of the disk at a particular point. As you stack these disks from the axis of rotation to the outer edge, the radius changes. This is where the function representing the boundary of the solid, \( R(x) \) or \( R(y) \) comes into play, describing the radius of the disks at each slice.

For the given exercise, by revolving the bounded region around the \(x\)-axis, the radius of each disk will be the vertical distance between the curve \( y = \sqrt{x} \) and the line \( y = -\frac{1}{2}x + 4 \). Thus, the area of the cross-sectional disk at any \(x\) is \( \pi[R(x)]^2 \) and the volume of the entire solid is found by integrating this area function along the \(x\)-axis from \(x = a\) to \(x = b\), the bounds of the region.

Definite Integral

A definite integral is a fundamental concept in calculus that represents the accumulation of quantities, which can be areas, volumes, or other physical quantities. It is written in the form \( \int_{a}^{b} f(x)\, dx \) and is interpreted as the signed area under the curve \( f(x) \) from \( x = a \) to \( x = b \).

The definite integral has many applications in mathematics and science, but when it relates to volume calculation using the disk method, the definite integral allows us to sum infinitesimally thin disks' volumes across the interval to find the solid's total volume. In our exercise, we evaluate the definite integral \( \int_{0}^{8} [(-1/2)x + 4 - \sqrt{x}]^2 dx \) which represents the sum of the areas of the disks, each with an area \( \pi[R(x)]^2 \) where \( R(x) = (-1/2)x + 4 - \sqrt{x} \) from \( x = 0 \) to \( x = 8 \). Evaluating this integral gives us the exact volume of the solid of revolution.

Volume Calculation

Volume calculation using calculus often involves integrating an area function to account for an accumulation of infinitely small elements over a region. In the context of solids of revolution, the goal is to determine the volume of a three-dimensional object created by rotating a two-dimensional shape about an axis. Calculating such volumes requires considering the shape and size of each differential element contributing to the solid.

In the exercise we're examining, the definitive measure of the solid's volume is attained by integrating the square of the function that represents the radius of our disks. This approach adopts the general formula for the volume of a cylinder, extended across an interval, taking into account the changing radius along the axis of rotation. Thus, the integral formula \( V = \pi \int_{a}^{b} [R(x)]^2\, dx \) is not just a symbolic expression; it quantifies the cumulation of the disk volumes as one single value, the volume of the solid of revolution you are determining.

Once the proper setup is done, calculating the volume comes down to evaluating the integral. This might involve algebraic manipulation, substitution, and applying the antiderivative rules of calculus. For simpler functions, the evaluation is straightforward, as seen in the provided solution where the integration limits and the function are clear-cut, which simplifies the computational process leading to the final volume.