Hello. We need help with a math lab. We use the platform wolfram mathematica and it is a mix of math and computer science. Z = integral sign.

Exercise 1. Compute the integral Z Z R (x + y) 5 dA, where R is the rectangle 1 x 2, 0 y 3.

Exercise 2. Compute the integral Z Z D e x+3y dA, where D is the (type II) region described by 0 y 1, 0 x 3 + y.

Exercise 3. Compute the area of the part of the ellipse x 2 + 3y 2 = 1 that lies in the positive quadrant. (Hint. A(D) = RR D dA.)

Exercise 4. Continuing with the previous D (lying between the hyperbola and the line), compute Z Z D (x + 2y) 3 dA. (Hint. With the computation of the intersection of the two curves, you can describe the domain as a Type I and as a Type II domain.)

Exercise 5. The average of a function f over a region D is defined as 1 A(D) Z Z D f(x, y)dA. Compute the average of 1 1 + x 2 + 2y 2 in the area of the unit circle (x 2 + y 2 1) that lies in the first quadrant

Exercise 6. Let D be the triangular region delimited by the lines x + y = 1, x 2y = 0, y = 1. Plot the three lines in a graph that shows the entire region. You might have to play with the limits for x and for y to get to see the entire figure. To help you with this, take 2 x 4 and dont specify the range (that is, dont use the option PlotRange at all.) Compute the three intersections of the pairs of lines using Solve. Finally, compute Z Z D (x y) 3 dA.

Exercise 7. Plot and find the exact volume of the solid that lies under the surface z = 1 4x 2y 2 and over the unit disk, i.e. the region inside the unit circle. Hint: To plot the solid, use the following command lines: Plot3D[1-4x^2y^2, {x, -1, 1}, {y,-1,1}, Filling->Bottom, RegionFunction->Function[{x,y}, x^2+y^2<1]]