calculus Flashcards
What is the washer method for finding volume of three dimensional shapes
The washer method is a variant of the disk method for using integration to find the volume of 3D objects. A function is rotated around the X axis which creates a circular shape.
The formula for a circle is pi x radius squared. Since the radius of the circle is the y value at any given point to turn the formula in a shape you square the function and multiply the result by pi.
In the washer method a second function carves out a hollow core within the shape. Calculate and integrate both of these functions then subtract the volume of the inner function from the outer function.
Explain the disk method for finding the volume of a cyclindrical object
The disk method is creating a shape by rotating a function around the x axis. The formula for area of a circle is πradius²Therefore to create an object from the function, the whole function must be squared and multiplied by pi.
The resulting 3D shape will have a disk shaped cross section at every point whose radius is given by the y value at that point (x value) in the function.
The volume of the object will be the sum of the volume of all of those infinitesimally thin disks.
Describe the shell method for finding the volume of 3D shapes.
In the disk and washer method functions are transformed into 3d shapes by rotating the function around the x (horizontal) axis.
In the shell method functions are turned into 3D shapes by rotation around the Y axis. The circumference of a circle is given by 2piRadius. The radius in this equation is the dx so to rotate it around the y axis multiply the derivative by 2pi X f(X) dX
Keep the constant 2pi outside for convenience
Multiply the function by X
Take the derivative of the above product
Integrate the new derivative
Evaluate for the upper and lower limits of integration be sure to include 2pi in the evaluation.
It is useful to imagine the resulting 3D shapes as being composed of a series of infinitely thin concentric cylinders. Like rings on a tree.
Explain the quotient method of taking derivatives
The quotient rule is really a variation of the product rule. It is used for rational functions when one function is divided by another.
F(x)/g(x) (f(x)g’(x)-f’(x)g(x))/g(x)^2
Top times the derivative of the bottom minus the bottom time the derivative of the top over the bottom squared.
