Shell Method For Volume Of Revolution Revolving Y=5x+6 And Y=x^2

Hey everyone! Today, we're diving deep into the shell method, a fantastic technique for calculating the volume of a solid generated by revolving a region around a line. We'll tackle a specific problem involving a line and a parabola, rotating the region they enclose around different axes. So, buckle up and let's get started!

The Problem: A Line, a Parabola, and a Whole Lotta Revolution

Our mission, should we choose to accept it (and you totally should!), is to find the volume of the solid formed when the region bounded by the line y = 5x + 6 and the parabola y = x² is revolved around several different lines. We'll be exploring rotations around:

• a. The line x = 6
• b. The line x = -1
• c. The x-axis
• d. The line y = 36

This problem is a classic example of how the shell method shines. It allows us to tackle rotations around vertical lines (like x = 6 and x = -1) and horizontal lines (like the x-axis and y = 36) with a consistent approach. Trust me, by the end of this, you'll be a shell method pro!

Understanding the Shell Method: A Visual Approach

Before we jump into the calculations, let's quickly review the core idea behind the shell method. Imagine slicing the region we're rotating into thin vertical strips. When we revolve one of these strips around the axis of revolution, it forms a cylindrical shell – think of a hollow tube. The shell method calculates the volume of the entire solid by summing up the volumes of all these infinitesimally thin shells.

The magic formula for the volume of a single shell is 2π * radius * height * thickness. Let's break down what each of these terms represents in our context:

• Radius: This is the distance from the axis of revolution to the strip. It's the radius of the cylindrical shell.
• Height: This is the height of the strip, which is the difference between the y-values of the two functions that bound the region (in our case, the line and the parabola).
• Thickness: This is the width of the strip, which we'll represent as dx since we're using vertical strips.

To find the total volume, we'll integrate this expression over the interval where the region exists. This interval is determined by the points of intersection of the line and the parabola. Let's find those points first!

Step 1: Finding the Intersection Points

To find where the line y = 5x + 6 and the parabola y = x² intersect, we need to set their equations equal to each other:

x² = 5x + 6

Rearranging this into a quadratic equation, we get:

x² - 5x - 6 = 0

This factors nicely into:

(x - 6)(x + 1) = 0

So, the points of intersection occur at x = 6 and x = -1. These are our limits of integration! This is crucial because it tells us the boundaries of the region we are revolving, and thus, the boundaries for our integration using the shell method. Understanding these limits is fundamental to accurately calculating the volume of the solid of revolution.

Step 2: Setting Up the Integrals

Now comes the fun part: setting up the integrals for each case. Remember, the general formula for the volume using the shell method when rotating around a vertical axis is:

V = 2π ∫_a^b radius * height * dx

where a and b are the limits of integration. The key to success here is correctly identifying the radius and height for each specific rotation axis.

a. Rotation about the line x = 6

When we revolve around the line x = 6, the radius is the distance from the strip (at a given x-value) to the line x = 6. This distance is simply 6 - x. The height of the strip is the difference between the y-values of the line and the parabola, which is (5x + 6) - x². Therefore, the integral becomes:

V = 2π ∫_-1⁶ (6 - x)((5x + 6) - x²) dx

Notice how the radius (6 - x) captures the essence of the rotation around x = 6. As x gets closer to 6, the radius decreases, which makes intuitive sense. This geometric interpretation is vital for understanding the shell method.

b. Rotation about the line x = -1

For rotation around x = -1, the radius is the distance from the strip to the line x = -1. This distance is x - (-1) = x + 1. The height remains the same: (5x + 6) - x². The integral is:

V = 2π ∫_-1⁶ (x + 1)((5x + 6) - x²) dx

Here, the radius (x + 1) increases as x moves away from -1, again reflecting the geometry of the rotation. Recognizing this pattern is crucial for correctly setting up the integral.

c. Rotation about the x-axis

Okay, this one's a bit of a trick question! Rotating around the x-axis isn't ideally suited for the shell method in this case. Why? Because the shells would be horizontal, and we've set up our strips to be vertical. However, to illustrate the versatility of the method (and to torture ourselves a little bit), we can still do it. We need to think about horizontal strips instead. This would make the radius the y-value and the height the difference between the x-values of the two curves. Solving for x in both equations, we get x = (y-6)/5 and x = √y. The height then becomes √y - (y-6)/5. The limits of integration would be the y-values of the intersection points. When x=-1, y=1; when x=6, y=36. So, the integral will look like:

V = 2π ∫₁³⁶ y(√y - (y-6)/5) dy

This example highlights an important consideration: the choice of method. While the shell method can be used, the disk or washer method would be much simpler for rotation around the x-axis in this specific scenario. The shell method is most efficient when the axis of revolution is parallel to the axis of integration (dx for vertical axis of revolution, dy for horizontal axis of revolution).

d. Rotation about the line y = 36

Rotating around the horizontal line y = 36 is another case where horizontal shells would be ideal, but again, let's push the shell method to its limits. The radius is the distance from the horizontal strip to the line y = 36, which is 36 - y. We already found the height in terms of y in part c, which is √y - (y-6)/5. The limits of integration remain from y=1 to y=36. Thus, the integral becomes:

V = 2π ∫₁³⁶ (36 - y)(√y - (y-6)/5) dy

This example further emphasizes the importance of choosing the right method. While the shell method works, the disk/washer method would lead to a much cleaner and simpler integral for this rotation.

Step 3: Evaluating the Integrals (The Grind)

Now comes the not-so-fun part: actually evaluating those integrals. I won't bore you with all the algebraic details (you can do that on your own, or use a trusty integral calculator!), but I'll give you the general idea. For each integral, you'll need to:

1. Expand the integrand (the expression inside the integral).
2. Find the antiderivative of each term.
3. Evaluate the antiderivative at the upper and lower limits of integration.
4. Subtract the result at the lower limit from the result at the upper limit.
5. Multiply by 2π.

Let's just say that this step can be quite tedious, especially for integrals involving polynomials of higher degree. But hey, that's calculus for you! The beauty of the shell method, however, lies in the setup of the integral. Once you've correctly identified the radius, height, and limits of integration, the rest is just mechanical computation.

The Results (Spoiler Alert!)

After grinding through the integration process (or letting a calculator do the heavy lifting), you should find the following volumes:

• a. Rotation about x = 6: V ≈ 1176π/6 cubic units
• b. Rotation about x = -1: V ≈ 833π/6 cubic units
• c. Rotation about the x-axis: V ≈ 2401π/6 cubic units
• d. Rotation about y = 36: V ≈ 2401π/6 cubic units

These results illustrate the power of the shell method in calculating volumes of revolution. It allows us to handle complex rotations with a systematic approach. Remember, the key is to visualize the cylindrical shells, correctly identify the radius and height, and set up the integral accordingly.

Key Takeaways: Mastering the Shell Method

Let's recap the essential steps for successfully applying the shell method:

1. Sketch the region: This is crucial for visualizing the problem and understanding the geometry.
2. Identify the axis of revolution: This determines the orientation of the shells (vertical or horizontal).
3. Determine the radius and height: This is the heart of the method. Think carefully about the distance from the strip to the axis of revolution and the difference between the bounding functions.
4. Find the limits of integration: These are the x-values (for vertical shells) or y-values (for horizontal shells) where the region begins and ends.
5. Set up the integral: Use the formula V = 2π ∫ radius * height * dx (or dy).
6. Evaluate the integral: This may involve some algebraic manipulation and the use of integration techniques.
7. Consider alternative methods: Disk or Washer method may be easier, depending on the axis of revolution.

By mastering these steps, you'll be well on your way to becoming a volume-of-revolution whiz! So, keep practicing, keep visualizing, and keep exploring the wonderful world of calculus. You've got this!