Problem 17 - Single Variable Calculus

Question

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=x^{2}+1, \quad x \geq 0, \quad y=5 ; \quad\) the \(y\) -axis

Step-by-Step Solution

Answer

The volume of the solid generated by revolving the region bounded by the graphs of the given equations is \(8\pi \) cubic units. The region is defined by the equation \(y = x^2 + 1\) (a parabola) and the line \(y = 5\) with \(x\geq 0\). We used the method of cylindrical shells with a thickness of \(\mathrm{d}x\), and the range of integration was \([0, 2]\). The height of the cylinder is \((4 - x^2)\), and the volume is calculated using the formula \(V = 2\pi \int_{a}^{b} x \cdot h(x) \, \mathrm{d}x \).

Step 1: Sketch the Region

First, we will sketch the region bounded by the given conditions. Our region is defined by the equation \(y = x^2 + 1\) (a parabola) and the line \(y = 5\) with \(x\geq 0\). On top of that, we should also sketch a representative rectangle that we will revolve around the y-axis to help in visualization. [asy] import graph; size(200); real xMin = -1.0, xMax = 2.0, yMin = 0.0, yMaxY = 6.0; real f (real x) { return x^2 + 1; } real g (real x) { return 5;} real h (real y) { return sqrt(y-1);} draw(graph(f, 0, 1.2), blue, linewidth(1)); draw((0,1)--(0,5), red, linewidth(1)); draw(graph(h, 1, 5), blue+dashed , linewidth(0.5)); draw(graph(g, -1, 2), green, linewidth(1)); draw((1,5)--(1,1.2), black, linewidth(0.5)); label("\(y=x^2+1\)", (1.4,4),fontsize(8), blue); label("\(y=5\)", (2,5.3), fontsize(8), green); label("\(x=0\)", (-0.4,3.8), fontsize(8), red); fill((0,1)--(0,5)--(1,5)--(1,2)--cycle, gray(0.7)); [/asy] Here, we have the parabola (blue), the line \(y=5\) (green), and the y-axis (red). The filled gray area represents the region we will revolve around the y-axis. The dashed parabola is just the reflection of the original parabola across the y-axis.

Step 2: Find the Thickness of Cylindrical Shells

We will now find the thickness of the cylindrical shells, which will be represented as \(\mathrm{d}x\). The reason is that as we revolve around the y-axis, each cylinder's radius will be a function of x and its height will be determined by the difference in the y values (\(5-y\) in this case). Hence, using \(\mathrm{d}x\) as the thickness is appropriate.

Step 3: Determine the Range of Integration

Our range of integration will cover the x-values the region covers as we move along the y-axis. To determine those, we need to find the intersection points of \(y = x^2 + 1\) and \(y = 5\). \(5=x^2+1\) \(x^2=4\) \(x=2, x=-2\) Since we are only concerned with \(x\geq 0\), we have the range of integration: \([0, 2]\).

Step 4: Calculate the Volume

Now we will apply the method of cylindrical shells to find the volume of the solid. The volume is given by the formula: \(V = 2\pi \int_{a}^{b} x \cdot h(x) \, \mathrm{d}x \) Where a and b are the limits of integration, x is the radius of the cylinder, and h(x) is the height of the cylinder at a given x value. In this case, \(a = 0\), \(b = 2\), and the height of the cylinder is \((5 - (x^2 + 1)) = (5 - x^2 - 1) = (4 - x^2)\). Plugging in the values: \[V = 2\pi \int_{0}^{2} x(4-x^2) \, \mathrm{d}x\] To find this integral, we will apply the power rule and the constant factor rule: \(V = 2\pi \left[ \int_{0}^{2} 4x \, \mathrm{d}x - \int_{0}^{2} x^3 \, \mathrm{d}x \right] \) \(V = 2\pi \left[ \frac{4}{2}x^2 \bigg|_0^2 - \frac{1}{4}x^4 \bigg|_0^2\right] \) \(V = 2\pi \left[ 8 - \frac{16}{4} \right] \) \(V = 2\pi (8 - 4) \) \[V = 8\pi \] The volume of the solid generated by revolving the region bounded by the graphs of the given equations is \(8\pi \) cubic units.

Key Concepts

Volume of Revolution

Imagine you have a shape, and you want to see how much space it takes up when it's revolved around a line. This is what the volume of revolution is all about. Using the method of cylindrical shells, we can easily calculate the volume of a solid of revolution. This method involves imagining the solid as being made up of many thin "shells," much like layers of an onion, that stack together to form the entire shape. When each shell is imagined, it's like revolving a thin rectangle, ensuring you cover the entire region. To find the total volume, you sum up the volumes of these thin cylindrical shells with an integral. This provides a handy and efficient way of calculating volumes for complex shapes by revolving them. Understanding the cylindrical shells method helps greatly in visualizing and calculating how spaces transform when they are rotated.

Integration Limits

Setting the correct integration limits is crucial when determining the volume of a solid using the cylindrical shells method. These limits are essentially the boundaries of the area you're planning to rotate. In our example, the area is formed by the intersection of a curve and a straight line, providing us with the limits. Solving the equations for intersection points gives you the start and end of your integral. For the specific problem given, the limits come from the set points where the parabola and horizontal line intersect, spanning from 0 to 2. By establishing these, you’ve mapped the "edges" of your solid so the integral knows where to start and stop calculating.

Parabolas

A parabola is a symmetrical, U-shaped graph. It is represented by the quadratic equation of the form: \( y = ax^2 + bx + c \)In this context, it's an essential component, forming one of the boundaries. The function provided, \( y = x^2 + 1 \), is a simple parabola opening upwards. It's important because it defines part of the region under consideration.Since the parabola intersects with a horizontal line, it plays a role in determining the edges of the shape for the volume calculation. In such dynamics, understanding how this curve contributes to the boundary shapes the foundation for further calculation.

Axis of Rotation

When a region is rotated around an axis, it's crucial to understand where that rotation happens. Here, the axis of rotation is the y-axis. This setup influences how you approach the problem, as it dictates the orientation of your cylindrical shells. Rotating around the y-axis means your shells will be vertical with respect to the x-axis. Each shell's radius will be determined by the x-coordinate, while the height is a function of the difference in y-values. Having clarity around your axis of rotation is essential, as it determines how the shells pile up and dictates aspects like the limits and dimensions of the volume you're calculating.