Calculating Volume Of Revolution Using Shell Method Parabola Rotation

In the realm of calculus, determining the volume of a solid of revolution stands as a fundamental concept with wide-ranging applications in physics, engineering, and computer graphics. This article delves into the intricacies of calculating the volume generated when a region in the first quadrant, bounded by a parabola $f(x) = -3x^2 + 18x + 81$, the x-axis ($y=0$), and the y-axis ($x=0$), is rotated about the y-axis. We will explore the shell method, a powerful technique for solving such problems, and arrive at the exact volume without resorting to decimal approximations. So, if you're ready to embark on a journey through the world of calculus and solidify your understanding of volumes of revolution, let's dive in.

At its core, this problem challenges us to visualize and quantify the three-dimensional shape formed when a two-dimensional region is spun around an axis. The region in question is confined to the first quadrant, a space where both x and y coordinates are positive. This region is defined by three boundaries: a parabola described by the equation $f(x) = -3x^2 + 18x + 81$, the vertical line $x = 0$ (the y-axis), and the horizontal line $y = 0$ (the x-axis). The act of rotating this region about the y-axis transforms it into a solid, and our mission is to precisely calculate the volume of this solid. To tackle this challenge, we'll employ the shell method, a calculus technique particularly well-suited for scenarios involving rotation about the y-axis. This method involves envisioning the solid as being composed of infinitesimally thin cylindrical shells, each with a volume that can be calculated. By summing the volumes of these shells through integration, we can determine the total volume of the solid of revolution. Before we delve into the calculations, it's crucial to grasp the geometry of the region and how it transforms upon rotation. The parabolic function plays a pivotal role, dictating the shape of the region and, consequently, the form of the solid. The points where the parabola intersects the x and y axes, along with its vertex, are key features that will influence our setup of the integral. The choice of the shell method stems from its effectiveness in handling rotations about vertical axes. It offers a direct way to relate the volume to the original function and the limits of integration, making it a powerful tool in our calculus arsenal. Now, with a clear understanding of the problem and our chosen approach, we're well-equipped to embark on the journey of calculating the exact volume of this captivating solid of revolution.

Choosing the Shell Method

The shell method emerges as the ideal technique for this specific problem due to the orientation of the axis of rotation (the y-axis) and the nature of the bounding function. To truly appreciate the elegance of the shell method, it's helpful to contrast it with the disk or washer method, another common approach for calculating volumes of revolution. The disk or washer method involves slicing the solid perpendicular to the axis of rotation, creating disks or washers whose areas are integrated to find the volume. However, in our case, if we were to use the disk or washer method, we would need to express x as a function of y, which would involve solving the quadratic equation for x. This can be a cumbersome process, potentially leading to complicated integrals. The shell method, on the other hand, bypasses this complexity. It involves slicing the solid parallel to the axis of rotation, forming cylindrical shells. Each shell has a circumference, a height, and a thickness, all of which can be expressed in terms of x, the variable we already have in our function $f(x)$. This direct relationship with the original function makes the shell method a more streamlined and efficient approach for this problem. The volume of each cylindrical shell is given by $2πrhΔx$, where r is the radius (the x-coordinate), h is the height (the function value $f(x)$), and $Δx$ is the thickness. By integrating this expression with respect to x over the appropriate interval, we sum the volumes of all the shells to obtain the total volume of the solid. The beauty of the shell method lies in its ability to avoid the need for inverting the function and dealing with potentially messy expressions. It leverages the existing function and the geometry of the problem to provide a clear and direct path to the solution. In our case, the shell method not only simplifies the calculations but also offers a more intuitive way to visualize the solid of revolution as a collection of concentric cylindrical shells, each contributing to the overall volume. So, with the shell method firmly in our sights, we're ready to embark on the next phase: setting up the integral that will unlock the exact volume of our solid.

Setting Up the Integral

The setup of the integral is a crucial step in applying the shell method, as it translates the geometric problem into a mathematical expression that we can solve. To construct the integral, we need to define the limits of integration and the integrand, which represents the volume of a single cylindrical shell. In our case, the region is bounded by the y-axis ($x=0$) on the left and by the parabola's intersection with the x-axis on the right. To find the right boundary, we need to determine the x-intercepts of the function $f(x) = -3x^2 + 18x + 81$. This involves solving the quadratic equation $-3x^2 + 18x + 81 = 0$ for x. We can simplify this equation by dividing by -3, resulting in $x^2 - 6x - 27 = 0$. Factoring this quadratic, we get $(x - 9)(x + 3) = 0$, which gives us two solutions: $x = 9$ and $x = -3$. Since we are in the first quadrant, we only consider the positive solution, $x = 9$. Thus, our limits of integration are from 0 to 9. The integrand represents the volume of a single cylindrical shell, which, as we discussed earlier, is given by $2πrhΔx$. In our context, the radius r is simply the x-coordinate, and the height h is the function value $f(x) = -3x^2 + 18x + 81$. Therefore, the integrand becomes $2πx(-3x^2 + 18x + 81)$. The complete integral is then expressed as: $V = ∫[0 to 9] 2πx(-3x^2 + 18x + 81) dx$ This integral encapsulates the essence of the problem, representing the summation of the volumes of infinitesimally thin cylindrical shells from $x = 0$ to $x = 9$. It's a testament to the power of calculus, allowing us to transform a geometric concept into a precise mathematical formulation. Before we proceed to evaluate the integral, it's worthwhile to pause and appreciate the elegance of this expression. It captures the shape of the parabola, the rotation about the y-axis, and the method of cylindrical shells, all within a single, compact equation. Now, with the integral set up, we're poised to embark on the final stage: the evaluation that will reveal the exact volume of our solid of revolution.

Evaluating the Integral

Evaluating the integral is the final step in our journey to determine the exact volume of the solid of revolution. The integral we established is: $V = ∫[0 to 9] 2πx(-3x^2 + 18x + 81) dx$ To begin, we can simplify the integrand by distributing the $2πx$ term: $V = 2π ∫[0 to 9] (-3x^3 + 18x^2 + 81x) dx$ Now, we can apply the power rule for integration, which states that $∫x^n dx = (x^(n+1))/(n+1) + C$, where C is the constant of integration. Applying this rule to each term in the integrand, we get: $V = 2π [(-3x^4)/4 + (18x^3)/3 + (81x^2)/2] [0 to 9]$ Simplifying the coefficients, we have: $V = 2π [(-3/4)x^4 + 6x^3 + (81/2)x^2] [0 to 9]$ Next, we evaluate the expression at the upper and lower limits of integration and subtract the results. When $x = 9$, we have: $(-3/4)(9^4) + 6(9^3) + (81/2)(9^2) = (-3/4)(6561) + 6(729) + (81/2)(81) = -4920.75 + 4374 + 3280.5 = 2733.75$ When $x = 0$, all terms become zero. Therefore, the volume is: $V = 2π(2733.75) = 2π(10935/4) = (10935π)/2$ Thus, the exact volume of the solid of revolution is $(10935π)/2$ cubic units. This result showcases the power of calculus in providing precise solutions to geometric problems. The journey from visualizing the solid to setting up the integral and finally evaluating it has culminated in a definitive answer, free from decimal approximations. The value $(10935π)/2$ not only represents the volume but also encapsulates the essence of the problem, reflecting the interplay between the parabola, the axis of rotation, and the method of cylindrical shells. With this final result, we've successfully navigated the realm of volumes of revolution and gained a deeper appreciation for the elegance and power of calculus.

Result

$$V = (10935π)/2$$

In conclusion, we have successfully calculated the volume of the solid of revolution generated by rotating the region bounded by the parabola $f(x) = -3x^2 + 18x + 81$, the x-axis, and the y-axis about the y-axis. By employing the shell method, we were able to set up and evaluate an integral that precisely captures the volume of this three-dimensional shape. The exact volume, determined to be $(10935π)/2$ cubic units, stands as a testament to the power of calculus in solving geometric problems. The journey through this problem has highlighted several key concepts in calculus. We've seen the importance of visualizing the solid of revolution and choosing the appropriate method for calculating its volume. The shell method, with its focus on cylindrical shells, proved to be particularly well-suited for this scenario, allowing us to avoid the complexities of inverting the function. The process of setting up the integral involved careful consideration of the limits of integration and the integrand, ensuring that the mathematical expression accurately reflected the geometric reality. The evaluation of the integral showcased the power rule for integration and the importance of meticulous calculations in arriving at the final answer. Beyond the specific solution, this problem serves as a valuable illustration of the broader applications of calculus in fields such as physics, engineering, and computer graphics. The ability to calculate volumes of revolution is essential in designing objects with specific shapes and properties, from machine parts to architectural structures. The principles we've explored here extend to a wide range of similar problems, solidifying our understanding of calculus and its role in the world around us. As we conclude this exploration, we carry with us not only the solution to this particular problem but also a deeper appreciation for the beauty and power of calculus in unlocking the secrets of geometry and beyond. This journey has reinforced the importance of visualization, strategic problem-solving, and careful execution in the pursuit of mathematical understanding. The volume of revolution problem, with its blend of geometry and calculus, stands as a testament to the interconnectedness of mathematical concepts and their relevance in the world we inhabit.