Washer Method
Definition
The washer method is a technique used to find the volume of a solid of revolution by integrating the difference between two functions, where one function is inside another. It involves slicing the solid into thin washers and summing their volumes.
Analogy
Imagine you have a stack of washers with different radii. By stacking them up, you can create a 3D object like a cylinder or a donut. The washer method works similarly, but instead of physical washers, we use infinitesimally thin washers to calculate the volume of irregular shapes.
Related terms
Volume: The measure of space occupied by a three-dimensional object. In calculus, it refers to finding the amount of space enclosed by curves or surfaces.
Solid of Revolution: A three-dimensional shape formed by rotating a curve or region around an axis. The washer method is commonly used to find the volume of solids of revolution.
Disk Method: Another technique for finding the volume of solids obtained by revolving curves around an axis. It involves slicing the solid into disks perpendicular to the axis and summing their volumes.
"Washer Method" appears in:
Study guides (2)
Practice Questions (16)
What does the width of each washer represent in the washer method?
Which region is being revolved around the x-axis in the washer method?
What determines the total volume of the solid in the washer method?
Which variable represents the width of each washer in the washer method?
What is the purpose of subtracting the volume of the smaller disk from the volume of the larger disk in the washer method?
Which axis of revolution would be used if the region is defined by two functions with y-values in the washer method?
Which quantity is multiplied by the width of each washer to calculate its volume in the washer method?
When finding the volume of a solid using the washer method, revolving around the y-axis, the bounds of integration are determined by:
Consider a region defined by the functions f(x) = x^2 and h(x) = 2x, revolved around the z-axis from x = 0 to x = 2. What is the volume of the solid formed using the Washer Method?
Consider a region defined by the functions g(y) = 2y and k(y) = 3y, revolved around the w-axis from y = 0 to y = 1. What is the volume of the solid formed using the Washer Method?
Consider a region defined by the functions f(x) = √x and h(x) = 2√x, revolved around the z-axis from x = 0 to x = 4. What is the volume of the solid formed using the Washer Method?
When using the Washer Method, what defines the inner and outer radii of each washer?
How is the volume of each washer calculated in the Washer Method?
Which step is necessary to find the total volume of the solid using the Washer Method?
When using the Washer Method around an axis that isn't the x- or y-axis, what determines the width of each washer?
What defines the boundaries of the region being revolved in the Washer Method?
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