In this tutorial we are going to learn how to find the surface area of revolution. What that means is that we are going to take a function y=f(x) and we are going to revolve this function around the y axis. Doing so we will get a surface. To calculate area of that surface we will use the formula in form of integral that will be given. Part of this formula might be recognized as something we saw in a previous lecture on arc length, and that is not a coincidence. We are going to see how are these two formulas linked.
remember this formula, it helps to recall that the square root part
comes from the arclength formula (which in turn comes from the
Pythagorean Theorem). The 2πf (x) part comes from the
circumference of a circle, 2πr, where the radius r is
the height of the curve f (x) that is revolving around the
that you must integrate the square root formula above. A common
mistake is to integrate the function itself, not the square root
formula. Of course, this would give you the area under the curve and
not the surface area of revolution.
similar mistake is to mix this up with the arclength formula, which
looks similar. Be careful which one you are asked for.
make the common algebraic mistake of thinking that
reduces to a + b! This is extremely wrong, and your
teacher will likely be merciless if you do it
common technique in problems of this nature is to make a
u-substitution for whatever is under the square root sign.
Then (hopefully) you can manipulate the expression outside the
square root (which comes from f (x)dx) into being the
du. However, you might have to do several steps of algebraic
manipulation, pulling factors in or out of the square root sign,
before this works.
the graph is being revolved around the y-axis, simply switch
the roles of x and y in the formula above. Be sure to
check carefully which one the problem is asking for.
its feasible, check that your answer makes sense. Unlike area
integrals, which can be negative if a curve goes below the x-axis,
surface area of revolution should always be positive!
Surface Area of Revolution
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.