The value of the double integral ∫ ∫R (10x + 10y) dA is 224.
To evaluate the double integral using the given transformation, we need
to determine the Jacobian of the transformation and transform the region of integration.
Jacobian of the Transformation
The Jacobian of the transformation is given by:
| J(u, v) | = | ∂x/∂u ∂y/∂u ∂x/∂v ∂y/∂v |
= | 1/5 0 1/5 -4/5 |
= 1/5
Transforming the Region of Integration
The original parallelogram R is defined by the vertices (−2, 8), (2, −8), (4, −6), and (0, 10). Substituting these values into the transformation equations, we get:
(−2, 8) → (0, 6)
(2, −8) → (4, −2)
(4, −6) → (8, −8)
(0, 10) → (0, 10)
The transformed parallelogram R' is defined by the vertices (0, 6), (4, −2), (8, −8), and (0, 10).
Evaluating the Integral
Using the transformed variables u and v, we can now evaluate the double integral:
∫ ∫R (10x + 10y) dA
∫ ∫R' (10(u + v)/5 + 10(v - 4u)/5) |J(u, v)| du dv
= ∫ ∫R' 10u + 10v |1/5| du dv
= ∫ ∫R' 2u + 2v du dv
= (u^2 + 2uv) |_0^8 |_0^{10}
= (64 + 160) - (0 + 0)
= 224
Therefore, the value of the double integral ∫ ∫R (10x + 10y) dA is 224.