Volumen De Sólido Con Método Del Disco: F(x) = Cos(x)
Let's dive into a classic calculus problem, guys! We're going to figure out the volume of a 3D shape formed by spinning a 2D area around the x-axis. This might sound intimidating, but we'll break it down using the method of disks. This method is super handy for finding volumes of solids of revolution. Think of it like slicing a loaf of bread – each slice is a disk, and we add up the volumes of all the disks to get the total volume. So, buckle up, and let's get started!
Understanding the Problem
First, let's make sure we all understand what we are trying to solve. Our mission, should we choose to accept it, is to find the volume of the solid that results when we rotate the region bounded by the curve f(x) = cos(x), the x-axis, and the vertical lines x = -π/2 and x = π/2 around the x-axis. This means we're taking the cosine curve between these two x-values and spinning it around the x-axis, creating a 3D shape. Visualizing this is key! Imagine the cosine curve waving between -π/2 and π/2, and then picture it spinning – it'll form a sort of football or spindle shape. Our job is to calculate the volume of this shape.
Key Elements:
- Function: f(x) = cos(x). This is the curve that defines the shape we're rotating.
- Bounds: x = -π/2 and x = π/2. These are the vertical lines that tell us where to start and stop rotating the curve.
- Axis of Rotation: The x-axis. This is the line we're spinning the region around.
- Method: Disk Method. This is the technique we'll use to calculate the volume.
Before we jump into the math, let's appreciate the beauty of calculus. We're taking a continuous curve and using the power of integration to find the volume of a complex 3D shape. Pretty cool, huh?
The Disk Method: A Quick Refresher
Okay, so before we tackle this specific problem, let's do a quick review of the disk method. At its heart, the disk method is all about slicing a 3D solid into infinitely thin disks. Each disk has a circular face, and its volume is just the area of that circle times its thickness. The magic of calculus lets us add up the volumes of all these infinitesimally thin disks to find the total volume of the solid.
Here's the breakdown:
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Imagine a Slice: Picture a thin vertical slice of the region we're rotating. When this slice is rotated around the x-axis, it forms a disk.
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Disk Radius: The radius of this disk is simply the function value f(x) at that particular x. In our case, it's cos(x).
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Disk Thickness: The thickness of the disk is an infinitesimally small change in x, which we call dx.
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Disk Volume: The volume of a single disk is the area of its circular face (πr²) times its thickness (dx). So, the volume of a single disk is π[f(x)]² dx.
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Integrate: To find the total volume, we add up the volumes of all the disks. This is where integration comes in. We integrate the disk volume formula over the interval of x-values that define our region. In mathematical terms, the volume V is given by:
V = ∫[a, b] π[f(x)]² dx
where a and b are the lower and upper bounds of our x-interval, respectively.
So, that's the disk method in a nutshell. We slice, we find the volume of each slice (disk), and we integrate to add up all the volumes. Now, let's apply this to our cosine function problem.
Applying the Disk Method to f(x) = cos(x)
Alright, guys, now for the fun part! We're going to apply the disk method to our specific problem where f(x) = cos(x), and our bounds are x = -π/2 and x = π/2. Let's walk through it step-by-step.
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Set up the Integral: Remember the formula V = ∫[a, b] π[f(x)]² dx? We need to plug in our values. Our function f(x) is cos(x), and our bounds a and b are -π/2 and π/2, respectively. So, our integral becomes:
V = ∫[-π/2, π/2] π[cos(x)]² dx
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Simplify the Integrand: Let's simplify the expression inside the integral. We have [cos(x)]², which is the same as cos²(x). So, our integral now looks like this:
V = ∫[-π/2, π/2] π cos²(x) dx
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Trigonometric Identity: Now, we need to deal with cos²(x). Integrating this directly can be tricky. Luckily, there's a handy trigonometric identity we can use: cos²(x) = (1 + cos(2x))/2. Let's substitute this into our integral:
V = ∫[-π/2, π/2] π [(1 + cos(2x))/2] dx
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Separate the Integral: To make things even easier, let's separate the integral into two parts:
V = π/2 ∫[-π/2, π/2] (1 + cos(2x)) dx
V = (π/2) [∫[-π/2, π/2] 1 dx + ∫[-π/2, π/2] cos(2x) dx]
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Evaluate the Integrals: Now, we can integrate each term separately. The integral of 1 with respect to x is simply x. The integral of cos(2x) is (1/2)sin(2x). So, we have:
V = (π/2) [[x](from -π/2 to π/2) + [(1/2)sin(2x)](from -π/2 to π/2)]
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Apply the Limits of Integration: Now, we need to plug in our limits of integration, π/2 and -π/2, and subtract. Let's do it:
V = (π/2) [[(π/2) - (-π/2)] + [(1/2)sin(2(π/2)) - (1/2)sin(2(-π/2))]]
V = (π/2) [[π] + [(1/2)sin(π) - (1/2)sin(-π)]]
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Simplify: Remember that sin(π) = 0 and sin(-π) = 0. So, the second term in the brackets becomes zero. We're left with:
V = (π/2) [π]
V = π²/2
The Grand Finale: The Volume!
And there you have it, folks! The volume of the solid generated by rotating the region bounded by f(x) = cos(x), the x-axis, and the lines x = -π/2 and x = π/2 around the x-axis is π²/2 cubic units.
That's a wrap on this problem! We successfully used the method of disks to find the volume of a solid of revolution. Remember, the key is to visualize the problem, understand the formula, and take it one step at a time. Whether you're tackling calculus problems or anything else in life, breaking it down makes all the difference. Keep practicing, and you'll become a volume-calculating pro in no time!
Key Takeaways:
- The disk method is a powerful tool for finding volumes of solids of revolution.
- Visualizing the solid and the disks is crucial for setting up the problem correctly.
- Trigonometric identities can be helpful in simplifying integrals.
- Carefully apply the limits of integration to get the correct answer.
So next time you encounter a solid of revolution, remember the disk method, and you'll be spinning your way to success!