14.1 Iterated Integrals The image on the left shows loop structures in the atmosphere of the Sun consisting of superheated ionised gas shaped by the powerful magnetic fields generated by the Sun. (Image, courtesy of NASA, was taken by the TRACE satellite). The Solar Magnetospheric Theory Group in St Andrews developes models for a diverse range of phenomena that occur on the Sun, as well as investigating the behaviour of the Earth's magnetosphere. More information can be found at:http://www-maths.mcs.st-andrews.ac.uk/images/sun.html

Example 1 Integrate with respect to y

Example 1 Integrate with respect to y

Example 2: Evaluate

Example 3 Use an iterated integral to represent the area of the rectangle shown (check with your knowledge of geometry)

Example 4 Use an iterated integral to find the area of the region bounded by the graphs of f(x) = sinx and g(x) = cosx for the region R

Example 4 solution

Example 5 Comparing different orders of integration. Find the area of the given region by using the integration order dx dy

Example 5 Solution 1

Example 5 continued Find the area of the region by using the integration order dy dx

Example 5 Solution 2