Question: Evaluate the double integral xy dA where R is the region bounded by the graphs of y= square root x, y= 1/2x, x=2, x=4
The first step is to sketch the graph, plotting the four different curves given, and to identify the region bounded by these curves.
(Please refer to the attachment - "Graph.png")
y = (square root of x)
represents a parabola which opens to the right.
[This is easy to note, since y = (sqrt)x means y^2 = x, that is, x = y^2: which means if the value of x is calculated, given values of y, we will get the same value for y and -y, since (-y)^2 = y^2. That is, x = y^2 will be symmetric about the x-axis, and obviously will take positive values, meaning the curve lie ...
This solution provides a detailed, step-wise response which depicts carefully how to evaluate the given double integral. Two attachments are also included in this solution, both .png files, in which one contains a sketch of the graph for this double integral and the other contains the integral.