David Little
Mathematics Department
Penn State University
Eberly College of Science
University Park, PA 16802
Office: 403 McAllister
Phone: (814) 865-3329
Fax: (814) 865-3735
e-mail:dlittle@psu.edu

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Area Bounded by a Polar Curve

The following applet approximates the area bounded by the curve r=r(t) in Polar Coordinates for a ≤ t ≤ b. Simply enter the function r(t) and the values a, b (in radians) and 0 ≤ n ≤ 1,000, the number of subintervals.

The values a and b can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change these values by using the up/down arrow keys or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

Here is a list of functions that can be used with this applet.

Use the above applet to:

  • approximate the area bounded by the cardiod, r(t) = 1 - sin(t).
  • approximate the area bounded by one loop of the curve r(t) = cos(2t).
  • approximate the area bounded by the inner loop of the curve r(t) = 1 + 2cos(t).
  • approximate the area bounded the curve r(t) = 1 + 2cos(t).
  • approximate the area bounded by the smaller loop of the curve r(t) = 1+2cos(4t).
  • approximate the area bounded by the larger loop of the curve r(t) = 1+2cos(4t)
  • consider the area bounded by curve r(t) = 2sin(t/2) for 0 ≤ t ≤ π. What is the exact value of this area. Can you explain why the approximation for this area for any value of n gives the exact value?
  • consider the area bounded by curve r(t) = 1+2cos(t) for 0 ≤ t ≤ π. What is the exact value of this area. Can you explain why the approximation for this area for any value of n>1 gives the exact value?

© 2005 David P. Little
Download this applet for off-line viewing (includes source code). The above applet uses the Java Math Expression Parser (JEP) developed by Singular Systems