Question

When the area under the graph of \(y(x)\) between \(x=a\) and \(x=b\) is rotated around the \(x\) axis, show that the volume of the solid formed is \(\int_{a}^{b} \pi y^{2} \mathrm{~d} x\)

Short answer

Question: Demonstrate that the volume of the solid formed when the area under the graph of a curve y(x) between x=a and x=b is rotated around the x-axis can be calculated using the integral \(\int_{a}^{b}\pi y^{2} \mathrm{~d} x\). Answer: To prove this, we consider a thin slice of the solid at a particular x value with thickness dx. The cross-section of the slice is a circle with radius equal to y(x). Calculating the volume of the slice using the cylinder formula, we get \(πy^2(x) \mathrm{~d}x\). By summing up the volumes of all slices between x=a and x=b using an integral, the volume of the solid is \(\pi \int_{a}^{b} y^{2}(x) \mathrm{~d} x\), which proves that the volume can be calculated using \(\int_{a}^{b}\pi y^{2} \mathrm{~d} x\).

Step-by-step solution

Consider a slice of the solid

Let's consider a thin slice of the solid at a particular x value, with thickness dx. Since the curve is rotated around the x-axis, the cross-section of the slice is a circle with radius equal to the value of y(x) at that x.

Calculate the volume of the slice

To find the volume of a slice, we will use the formula for the volume of a cylinder: V = πr²h. In our case, the radius of the cylinder r is equal to y(x), and the height of the cylinder h is the thickness dx of the slice. Plugging r and h into the formula results in: Volume of a slice = \(πy^2(x) \mathrm{~d}x\).

Add up the volumes of all slices

Now, let's sum up all the volumes of the slices from x=a to x=b. To do this, we will use an integral. Volume of solid = \(\int_{a}^{b} (\pi y^{2}(x) \mathrm{~d} x)\).

Show that the volume of the solid is \(\int_{a}^{b} \pi y^{2} \mathrm{~d} x\)

Since the constant π can be factored out of the integral, the expression simplifies to: Volume of solid = \(\pi \int_{a}^{b} y^{2}(x) \mathrm{~d} x\). And that demonstrates that the volume of the solid formed when the area under the graph of y(x) between x=a and x=b is rotated around the x-axis is \(\int_{a}^{b}\pi y^{2} \mathrm{~d} x\).

Key concepts

Solid of Revolution

Understanding the concept of a 'solid of revolution' is key to visualizing how these shapes are created and computed in mathematics. When we talk about a solid of revolution, we are describing a three-dimensional object obtained by rotating a two-dimensional shape, or area, around an axis. The result is a symmetric object such as a donut shape (torus), a cylinder, or a sphere, depending on the original shape.

Imagine you have a curve on a graph, like the one described by the function y(x), and you revolve this curve around a line, usually one of the coordinate axes. The area swept by the curve generates the surface of a solid. For instance, if you take the graph of y = sqrt(x) between x=0 and x=1 and rotate it around the x-axis, you get a cone. Calculating the volume of this cone involves taking into consideration every tiny slice of the solid that can be thought of as a disc or a cylinder, finding the volume of each, and then adding them all together.

Integral Calculus

Integral calculus is a branch of mathematics that deals with the accumulation of quantities and the areas under and between curves. When it comes to finding the volume of a solid of revolution, integral calculus allows us to sum an infinite number of infinitesimally thin slices (discs or cylindrical shells) which comprise our three-dimensional object.

The basic idea is that if we can calculate the volume of one small slice, we can integrate that volume over the range we're interested in to find the total volume. It's like finding the total area under a curve, but in three dimensions. In formal terms, an integral \text{\( \int_{a}^{b} f(x) \, \mathrm{d}x \)} represents the accumulation of the quantities given by f(x) as x goes from a to b. The beauty of integral calculus lies in its ability to handle this accumulation process, no matter how complex the shape or how many slices are involved.

Area Under a Curve

Finding the Volume via Slicing

We often encounter the task of finding the area under a curve in both calculus and real-world applications. This concept serves as a foundation for integrating to find volumes. The area under a curve is the region enclosed between the curve and the horizontal axis, from one vertical boundary to another. Computing this area can tell you things like the distance traveled over time if you’re looking at a velocity curve, or in our case, help in finding the volume of a solid of revolution.

To find the volume of the solid of revolution, we imagine slicing the solid perpendicular to the axis of rotation. Each slice is a disk, and the area of that disk is the area of the circle, which is \( \pi r^2 \). Since the radius of each disk in this context is given by the y value of our function at a specific x, we replace r with y(x). As we 'add up' (integrate) all these infinitesimal areas from x=a to x=b, we obtain the volume of the solid.

Cylindrical Shells Method

Alternatives to Discs: Cylindrical Shells

Another method for finding the volume of solids of revolution is the 'cylindrical shells' method, useful especially when the solid is rotating around an axis outside of the shape. This approach involves imagining the object as made up of concentric cylindrical shells, rather than disks.

Each shell's volume is found by considering its circumference (2\pi r), height (h), and thickness (dx or dy), resulting in a volume \(2\pi r h \, \mathrm{d}x\) or \(2\pi r h \, \mathrm{d}y\), depending on the axis of rotation. By integrating this expression over the given range, we calculate the total volume of the solid.

This method is often more convenient when dealing with the area to be rotated around the y-axis, for example, because it simplifies the radius and height in terms of x or y, instead of translating everything to the other variable, which might be required in the disk method.