Volume Of Solid Of Revolution Step-by-Step Guide

Hey guys! Ever wondered how to find the volume of a funky 3D shape created by spinning a 2D area around an axis? It might sound intimidating, but with a little calculus magic, it's totally doable! Today, we're diving into a classic problem: finding the volume of the solid formed when we revolve the region bounded by the curve y = 2√(sin x), the x-axis (y = 0), and the vertical lines x = π/4 and x = π/2, around the x-axis. Buckle up, because we're going on a mathematical adventure!

1. Understanding the Problem: Visualizing the Solid

Before we jump into the calculations, let's get a good grasp of what we're actually trying to find. Imagine the graph of y = 2√(sin x). It's a wave-like curve that starts at (0, 0) and oscillates up and down. We're interested in the portion of this curve between x = π/4 and x = π/2. This section, along with the x-axis and the vertical lines, encloses a region in the xy-plane. Now, picture spinning this entire region around the x-axis. What do you get? You'll end up with a three-dimensional solid, kind of like a distorted football or a vase. Our mission is to calculate the volume of this solid.

The key here is to visualize the solid of revolution. Think of it as being made up of an infinite number of infinitesimally thin disks stacked next to each other along the x-axis. Each disk is formed by rotating a tiny vertical strip of the region around the x-axis. The thickness of each disk is dx, and the radius of each disk is given by the function y = 2√(sin x) at that particular x-value. So, the area of each disk is π(radius)² = π(2√(sin x))² = 4π sin x. To find the volume of the entire solid, we need to sum up the volumes of all these infinitesimally thin disks. And guess what? That's exactly what integration is for! We will use definite integral to evaluate the volume of solid revolution formed. We'll be integrating the area of the disk with respect to x, from the lower limit x = π/4 to the upper limit x = π/2. This method is often referred to as the disk method in calculus. Visualizing this process is crucial for understanding why the formula we're about to use works. Imagine slicing the solid into many thin disks; each disk's volume contributes to the total volume of the solid. The integral allows us to add up these infinitely many small volumes to get the precise volume of the entire shape.

2. Setting Up the Integral: The Disk Method

The method we'll use to calculate this volume is called the disk method. It's a powerful technique in calculus for finding the volume of solids of revolution. The disk method relies on the idea that we can approximate the solid as a stack of infinitesimally thin disks, as we discussed earlier. The volume of each disk is given by πr²h, where r is the radius of the disk and h is its thickness. In our case, the radius r is the distance from the x-axis to the curve y = 2√(sin x), which is simply the value of the function y itself. The thickness h is an infinitesimally small change in x, which we denote as dx. To find the total volume, we integrate the volume of these disks over the interval of x-values that define our region, from x = π/4 to x = π/2.

So, the formula for the volume V of the solid of revolution using the disk method is:

V = ∫[from a to b] π[f(x)]² dx

Where:

• a and b are the limits of integration (in our case, π/4 and π/2).
• f(x) is the function that defines the radius of the disk (in our case, 2√(sin x)).

Now, let's plug in our specific values. Our function f(x) is 2√(sin x), so [f(x)]² is (2√(sin x))² = 4 sin x. Our limits of integration are a = π/4 and b = π/2. Therefore, our integral becomes:

V = ∫[from π/4 to π/2] π(4 sin x) dx

We can pull the constant 4π out of the integral, which simplifies things a bit:

V = 4π ∫[from π/4 to π/2] sin x dx

Now, we're ready to tackle the integration itself! The setup is often the trickiest part, so if you've made it this far, you're well on your way to solving the problem. Remember, the disk method is a powerful tool for finding volumes of solids of revolution. Understanding the concept of summing up infinitesimally thin disks is key to mastering this technique. The formula for the volume is derived from the basic principle of integrating the area of the disks along the axis of revolution. By squaring the function f(x), we are essentially calculating the area of the circular cross-section at each point along the x-axis. Multiplying this area by dx gives us the infinitesimal volume of a single disk. The definite integral then sums up all these infinitesimal volumes over the specified interval, giving us the total volume of the solid.

3. Evaluating the Integral: Calculus Time!

Alright, guys, it's time to put our calculus skills to the test! We've got our integral set up: V = 4π ∫[from π/4 to π/2] sin x dx. Now we need to find the antiderivative of sin x. Do you remember what it is? That's right, the antiderivative of sin x is -cos x. So, we can rewrite our expression as:

V = 4π [-cos x] [from π/4 to π/2]

Now we need to evaluate this expression at our limits of integration. We'll plug in x = π/2 and x = π/4, and then subtract the second value from the first:

V = 4π [(-cos(π/2)) - (-cos(π/4))]

Do you remember the values of cos(π/2) and cos(π/4)? cos(π/2) is 0, and cos(π/4) is √2/2. Let's plug those in:

V = 4π [( -0) - (-√2/2)]

Simplifying, we get:

V = 4π (√2/2)

And finally:

V = 2π√2

So, the volume of the solid generated by revolving the region around the x-axis is 2π√2 cubic units! That's a pretty cool result, isn't it? We used the power of calculus to find the volume of a complex 3D shape. The process of evaluating the integral involves finding the antiderivative of the function being integrated and then applying the Fundamental Theorem of Calculus. This theorem tells us that the definite integral of a function between two limits is equal to the difference in the antiderivative evaluated at those limits. In our case, we found the antiderivative of sin x to be -cos x. We then evaluated -cos x at the upper limit (π/2) and the lower limit (π/4), and subtracted the second value from the first. This gave us the numerical value of the definite integral, which, when multiplied by the constant 4π, gave us the final volume of the solid. Remember to always include the units in your final answer when dealing with real-world applications. In this case, since we are calculating volume, the units are cubic units. This indicates that we are measuring the amount of three-dimensional space occupied by the solid.

4. The Answer and Its Significance

We've reached the end of our journey, guys! We've successfully calculated the volume of the solid of revolution. The answer is 2π√2 cubic units. This result is more than just a number; it represents the amount of space enclosed by the three-dimensional shape we created by revolving the region around the x-axis. Think about it – we took a 2D region and transformed it into a 3D object, and calculus allowed us to precisely measure its volume!

This type of calculation has tons of applications in the real world. Engineers use these techniques to design everything from tanks and containers to car parts and airplane fuselages. Architects use them to create unique and interesting building designs. Even medical professionals use them to model organs and other body parts. The ability to calculate volumes of complex shapes is a fundamental skill in many fields, and the disk method is a powerful tool for doing so.

The significance of the answer lies in its practical applications. Many real-world objects have shapes that can be modeled as solids of revolution. For example, a vase, a bowl, or even a rocket nozzle can be thought of as a shape formed by rotating a curve around an axis. By using the disk method or similar techniques, engineers and designers can calculate the volume of these objects, which is crucial for determining the amount of material needed to construct them, their weight, their capacity, and other important properties. In the context of mathematics, this problem demonstrates the power of calculus to solve geometric problems. It shows how the concepts of integration and the disk method can be used to find the volume of complex shapes that would be difficult or impossible to calculate using traditional geometric formulas. The process of setting up the integral, evaluating it, and interpreting the result reinforces the fundamental principles of calculus and provides a deeper understanding of its applications. Furthermore, this problem highlights the importance of visualization in mathematics. Being able to visualize the solid of revolution and understand how it is formed is crucial for setting up the correct integral and interpreting the result. This skill is essential for problem-solving in many areas of mathematics and science.

So, there you have it! We've conquered the challenge of finding the volume of a solid of revolution. I hope this step-by-step guide has made the process clear and understandable. Remember, guys, calculus might seem tough at first, but with practice and a solid understanding of the concepts, you can tackle even the most complex problems. Keep exploring, keep learning, and keep those mathematical gears turning!