Finding Volume Of Solid Of Revolution Y² = X - 2 And Y - 3 = 0

Have you ever wondered how to calculate the volume of a three-dimensional shape formed by rotating a two-dimensional region around an axis? This is a common problem in calculus, and it's often encountered in various fields like engineering and physics. In this comprehensive guide, we'll delve into the method of finding the volume of a solid generated by revolving the region bounded by the curves y² = x - 2 and y - 3 = 0 around the y-axis. So, buckle up, guys, and let's embark on this mathematical journey!

Understanding the Problem

Before we dive into the calculations, let's first grasp the essence of the problem. We are given two equations: y² = x - 2 and y - 3 = 0. These equations represent curves in the xy-plane. The first equation, y² = x - 2, represents a parabola opening to the right, with its vertex at (2, 0). The second equation, y - 3 = 0, simplifies to y = 3, which is a horizontal line. The region bounded by these two curves is the area enclosed between the parabola and the line.

Now, imagine rotating this bounded region around the y-axis. This rotation will generate a three-dimensional solid. Our mission is to determine the volume of this solid. There are several methods to tackle this problem, but we'll focus on the method of disks (or washers), which is particularly well-suited for this scenario. This method involves slicing the solid into infinitesimally thin disks perpendicular to the axis of rotation (in this case, the y-axis), calculating the volume of each disk, and then summing up these volumes using integration. Think of it like slicing a loaf of bread into very thin pieces and adding up the volume of each slice.

To effectively use the disk method, it's crucial to visualize the solid of revolution. Imagine the parabola and the line rotating around the y-axis. The resulting solid will have a hollow center due to the parabolic shape. This is why we'll actually be using the washer method, which is a variation of the disk method that accounts for the hollow center. The washers are essentially disks with a hole in the middle.

So, the first step in our journey is to rewrite the equation of the parabola in terms of y. Solving y² = x - 2 for x, we get x = y² + 2. This form will be more convenient for our calculations as we'll be integrating with respect to y. Remember, guys, rewriting equations to suit our needs is a common and crucial skill in problem-solving!

Setting Up the Integral

Now that we understand the problem and have the equations in the appropriate form, we're ready to set up the integral. This is where the magic of calculus truly shines! The washer method relies on the idea that the volume of each infinitesimally thin washer is approximately equal to π(R² - r²)dy, where R is the outer radius, r is the inner radius, and dy is the thickness of the washer. The outer radius is the distance from the y-axis to the outer curve, and the inner radius is the distance from the y-axis to the inner curve. In our case, the outer radius is given by x = y² + 2 (the parabola), and the inner radius is 0 (since the region is bounded by the y-axis on the left). The thickness of the washer is dy because we're integrating with respect to y.

Before we can set up the definite integral, we need to determine the limits of integration. These limits will be the y-values where the parabola and the line intersect. To find these points, we set the equations equal to each other: y² + 2 = x and y = 3. Since we already have y = 3 from the line equation, we can substitute this into the parabola equation (once solved for x) to find the x-coordinate of the intersection point: x = (3)² + 2 = 11. So, the point of intersection is (11, 3). However, we only need the y-values for our limits of integration. To find the other limit, we need to find where the parabola intersects the y-axis. This occurs when x = 0. Plugging x = 0 into the parabola equation, we get y² = 0 - 2, which simplifies to y² = -2. Since we can't have a real solution for y² = -2, this means the parabola doesn't actually intersect the y-axis. Instead, we need to find the lower limit of integration by considering the vertex of the parabola. The vertex occurs at x = 2, which corresponds to y² = 0, so y = 0. Therefore, our limits of integration are y = 0 and y = 3.

Now we have all the pieces of the puzzle! We can set up the definite integral to calculate the volume of the solid:

Volume = ∫[from 0 to 3] π((y² + 2)² - 0²) dy

This integral represents the sum of the volumes of all the infinitesimally thin washers from y = 0 to y = 3. The expression inside the integral, π((y² + 2)² - 0²), represents the area of a single washer. Guys, remember that the power of calculus lies in its ability to break down complex problems into smaller, manageable parts and then reassemble them to find the solution!

Evaluating the Integral

With the integral set up, the next step is to evaluate it. This involves finding the antiderivative of the integrand and then applying the limits of integration. Let's break it down. First, we expand the square inside the integral:

Volume = ∫[from 0 to 3] π(y⁴ + 4y² + 4) dy

Now, we can find the antiderivative of each term. Remember the power rule for integration: ∫xⁿ dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule, we get:

Volume = π [(y⁵/5) + (4y³/3) + 4y] evaluated from 0 to 3

Now, we evaluate the antiderivative at the upper and lower limits of integration and subtract the results. This is the fundamental theorem of calculus in action! Plugging in y = 3, we get:

π [(3⁵/5) + (4(3)³/3) + 4(3)] = π [(243/5) + 36 + 12] = π [(243/5) + 48]

Plugging in y = 0, we get:

π [(0⁵/5) + (4(0)³/3) + 4(0)] = 0

Subtracting the value at the lower limit from the value at the upper limit, we get:

Volume = π [(243/5) + 48] - 0 = π [(243/5) + (240/5)] = π (483/5)

Therefore, the volume of the solid is (483π)/5 cubic units. Woohoo! We did it, guys! We successfully calculated the volume of the solid of revolution.

Alternative Methods and Considerations

While we used the washer method in this example, there's another popular technique called the shell method that can also be used to find volumes of solids of revolution. The shell method involves slicing the solid into cylindrical shells parallel to the axis of rotation. The choice between the disk/washer method and the shell method often depends on the specific problem and which method leads to a simpler integral. In some cases, one method might be significantly easier to apply than the other.

Furthermore, it's important to consider the axis of rotation. If we were to rotate the region around the x-axis instead of the y-axis, we would need to adjust our setup accordingly. The disks/washers would be perpendicular to the x-axis, and we would integrate with respect to x. Similarly, the shell method would involve cylindrical shells parallel to the x-axis.

Guys, remember that practice makes perfect! The more you work through these types of problems, the more comfortable you'll become with the different methods and considerations.

Conclusion

Finding the volume of a solid of revolution can seem daunting at first, but by breaking down the problem into smaller steps and applying the power of calculus, it becomes a manageable and even enjoyable task. We've explored the method of washers, which involves slicing the solid into thin washers and summing their volumes using integration. We've also discussed the shell method as an alternative approach and highlighted the importance of considering the axis of rotation.

So, next time you encounter a solid of revolution, don't be intimidated! Remember the steps we've discussed, practice your integration skills, and you'll be well on your way to finding the volume. Keep exploring the fascinating world of calculus, guys, and you'll discover even more amazing applications of these powerful tools! Remember, the key is to understand the concepts, visualize the problem, and approach it systematically. Happy calculating!