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

Solution Preview

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")

The equation
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 ...

Solution Summary

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.

... You may have to solve the equations numerically ... Solution. ...Solving above equations, we get the following solutions x1 = 5+5 3 , y1 = 5−5 3 , z1 = 0, λ1 = −1 ...

... This solution utilizes double integration in polar ... Use double integration in polar coordinates to find the ...Solve by double integration in polar coordinates. ...

...Solve: ... The 12-pages solution contains detailed explanation how to solve the homogeneous heat equation on a 2D rectangle with mixed boundary conditions. ...

... (1.34) And a general solution is a ... of a sum is a sum of integrals: (1.46) Since the integration limits are constants, we can write the double integral as a ...

... Finding the volume of a solid region by using double integration is a standard problem in multi-variable calculus. This solution demonstrates the method by ...

... a. Solve the initial value problem y' = x^4 + x^2 + 1 / y^2 , y(0) = 2 ... The particular solution is: ... b. How long will it take for the population to double? ...

... 7. A) 9.6 B) 3.2 C) 8.0 D) No solution. 8. A) B) 2 C) e -x sin(x) D). ... 10. Which of the following is not a property of double integrals? A) B) C) D). 11. ...

... The integration limits are: The integral is then. The solution shows how to find the volume of solid using double and triple integrals step-by-step. ...

... clock_t endTime = clock(); double secs = GetTimeEllapsed ... base on time are examined in the solution. ... a program for multidimensional integration that employs ...