Question

Let f(x, y) = (x + y)e x 2−y 2 and R be the rectangle enclosed by the lines x − y = 0, x − y = 2, x + y = 0, and x + y = 3.

Answer 1

Answer:

(1/4)*(e⁶ - 7)

Step-by-step explanation:

a) Given

x − y = 0  if  x = 0  ⇒  y = 0

x − y = 2  if  x = 0  ⇒  y = -2;  if  y = 0   ⇒  x = 2

x + y = 0  if  x = 0   ⇒  y = 0

x + y = 3  if  x = 0  ⇒  y = 3;  if  y = 0   ⇒  x = 3

then we show the region R in the pics 1 and 2.

b) We make the change of variables as follows

u = x + y

v= x - y

If

x - y = 0  ⇒ v = 0

x − y = 2  ⇒ v = 2

x + y = 0  ⇒ u = 0

x + y = 3  ⇒ u = 3

Where u is the horizontal axis and v is the vertical axis, the new region S is shown in the pic 3.

c) We evaluate ∫∫R (x + y)*e∧(x² - y²)dA

The procedure is shown in the pic 4, where we have to calculate the Jacobian in order to use it to get the answer.