Volumes of Revolution

#Calc-2#Math#Integrals A Volume of Revolution is created by rotating a curve around an axis from one point to another.

Finding the Volume §

Disk/Washer Method §

The disk washer method, pictured above, involves finding the volume of multiple small disks and then integrating to accumulate them.

Formula ${\int }_a^b[\pi (outercurve{)}^2-\pi (innercurve{)}^2]dx$ You can also integrate in y, just be careful that you are actually using disks/washers.

Importantly, when using the disk/washer method, your rectangles should be perpendicular to the axis of revolution Another picture description

Shell Method §

In the Shell Method, you find the volume of many cylinders and integrate to accumulate those.

Explanation

The important thing with the shell method is that the rectangles end up parallel to the axis of revolution

Formula ${\int }_a^{b}2\pi radius\ast [heightorlengthoftherectangle]dx$ the radius is the distance the rectangle is from the axis of revolution (usually ends up being just x)

Surface Area of a Volume of Revolution §

FORMULA: ${\int }_a^{b}2\pi f(x)\sqrt{1+[f^{\prime }(x){]}^2}dx$ It is nearly identical to the length of a curve formula

Process is same as every other type of integration, find surface area of a small strip, integrate to accumulate. Here’s the drawn out version.