Amy has a master's degree in secondary education and has taught math at a public charter high school.
In this lesson, you will learn how the washer method makes your calculus life a little easier by giving you an easy formula and procedure for finding the volume of various shapes created by simple (and not-so-simple) functions.
The Washer Method Defined
The washer method uses the shape of a washer to help us find the volume of a shape created by rotating two functions around the x-axis. A washer is a little piece of hardware that is used in construction and building projects. It is made out of metal and is in the shape of a flat doughnut.
Let's see how this washer relates to finding our volume using the washer method. The washer method is used when you have two functions where you want to find the volume between the functions. Let's start with a couple of simple functions: y = 1 and y = 2. This gives us two horizontal lines.
Now we want to find the volume between these two lines when we rotate them around the x-axis. What happens when we rotate each line around the x-axis? We get a circle from every point on the line. If we slice the resultant shape, the space between the two rotated lines will look like a washer.
Can you see the washer shape in this illustration? No matter what kind of functions you have, the shape you get when you slice them will always be a washer of some size. Also, you will always have an outside function and an inside function. In this case, y = 2 is our outside function, and y = 1 is our inside function.
This washer shape helps us to find the formula for the volume. We know we can find the volume of this 3-D shape by integrating from a starting point to an ending point. The formula we need will provide us the area of each slice. Well, a washer is quite simply two circles of different size. You have a smaller circle inside a larger circle. The radius of each circle is provided by our functions.
The outer function is our outer radius, and the inside function is our inner radius. The formula for the area of a circle is pi * r^2. So the area of the washer is pi * R^2 - pi * r^2, where the R is the outer radius and r is the inner radius. Plugging the functions into our respective radii will give us a formula for the washer method we can use. Let's see what happens when we put this formula to use.
Washer Method with Lines
We will start with our first example with the two lines y = 1 and y = 2. Since these two lines go on indefinitely, we need to specify a starting point and an ending point. Our starting point will be x = 0, and our ending point will be x = 2.
Let's go through the process of evaluating the definite integral that is produced after plugging in all of the information into the formula. We've already established that our outer function is y = 2 and our inner function is y = 1. Let's plug this information into the formula and integrate.
So what we did was plug in our outer function and inner function where they belong and integrated it from the starting point to the ending point. After plugging in the functions, we were actually able to simplify the integral. At this point, we took the definite integral and evaluated it to get an answer of 6 pi. In calculus, pi is usually left as part of the answer and not multiplied out.
Washer Method with Curves
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Let's see what happens when we have functions that have curves in them. Our two functions are y = x^2 and y = x^4. Looking at the graph, it looks like they intersect each other. We want to find the volume of the rotated functions between 0 and the first intersection on the positive side of the x-axis. How do we find out where they intersect? We set the functions equal to each other and solve for x.
The graph intersects at 0, 1, and -1. 0 is the starting point, so the next point of intersection is 1. We need to move all our variables to the same side before solving so we do not miss any solutions (if we had divided by x^2 to begin with, we would have lost the 0 solution).
Now we have a starting point of 0 and an ending point of 1. Let's plug our functions into the formula for the washer method. Our outer function is y = x^2 because that is the one that is on top. Our inner function is y = x^4. Let's see what our solution is.
If our functions intersect each other, we have to set them equal to each other to find where they intersect. When we have two lines that don't intersect, we didn't have to do that. But both scenarios are very similar and follow the same pattern of solving. We identify our outer and inner functions and plug those into the formula, and then evaluate the definite integral over our region to find our volume.
We've looked at using the washer method with two lines and two curves. As you have seen, the method is the same, only the function changes. You can use the washer method for any two functions that you have.
Let's review. The washer method is a fairly straightforward method for finding the volume between two functions that are rotated around the x-axis. The formula involves the area of a circle and is easy to use. If two functions intersect each other, we need to find where they intersect by setting them equal to each other and solving.
Key Words and Formulas
Washer method: a method for finding the volume between two functions that are rotated around the x-axis
Area of a circle: pi*r^2
Area of a washer: pi*R^2 - pi*r^2 where R = outer radius and r = inner radius
Upon completion, display your ability to:
Demonstrate the washer method using straight and curved lines
Write the formulas for the volume and area of shapes
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