Use a double integral to find the area of the region bounded by the graph at the right, function given below.

Click on the picture to see another plot with terminal sides of angles theta drawn in blue extending a distance r from the origin (pole).  Click here to see the graph on "polar graph paper".

Use a double integral to find the area of the region indicated on the right, function given below.

Click here to see the graph on "polar graph paper".

Area of a Rose Petal

Find the area of one petal of the graph of

Notice the limits of integration.

Click on the picture at the right to enlarge.  Larger     QTanimLg    QTanimSm

Micromedia Audio/Video Including An Excel Worksheet (How To)

Excel Worksheet     Calculating Some Points

Wallis's Formulas are on page 536 in LHE 8th ed., page 538 in LE 9th ed.

Click here to see an animated point moving along the curve.  Theta is t in the animation and is taking on values from zero to pi.  QT

Click here to see the graph on "polar graph paper".

The double integral would represent the volume of the region between the two surfaces shown above.  That would be the figure indicated in this DPGraphPicture.  Here is a DPGraphPicture showing the intersecting surfaces.

Here is a DPGraph picture of the surface z = x² graphed over the region of integration done using cylindrical coordinates and here is the same surface along with the cylinder

x² + y² = 4y graphed transparently making use of cylindrical coordinates.  The graphs have actually been shifted in order to be able to draw them this way--that is why "Box" has been turned off.

Maple Worksheet