double integration?

Question

if R = [-1, 3] [0, 2], use a Riemann sum with m = 4, n = 2 to estimate the value of the following. Take the sample points to be the upper left corners of the subrectangles.

int int_R (y^2 - 2x^2) dA

Answer 1

Make a rectangular grid with 1x1 square-base cells whose heights are given by y^2 - 2x^2. The Riemann sum is the total volume of all the cells, which is just the sum of the heights because the areas of the bases are all 1.

x-coord of left edges of rectangles: -1,0,+1,+2
y-coord of top/upper edges: 1, 2

Eight upper-left corners: (-1,1), (-1,2), etc. ....

Here's the calculation I did using matlab. It should be easy to understand even if you don't know matlab.

>> x = -1:2

x = -1, 0 , 1, 2

>> y = 1:2

y = 1 , 2

>> sum = 0;

>> for i=1:4
for j = 1:2
sum = sum + y(j)^2 - 2*x(i)^2
end;
end;

>> sum

sum = -4 (the answer - approximate value of the integral)

Answer 2

Riemann Sum Matlab

Answer 3

assume you elect for to make certain the area between the curves y = x^2 and y = x^3 from x = a million to x= 3. that's complete via the double fundamental Int (x=0 to a million) Int (y = x^2 to x^3) dy dx