Question

To evaluate the following​ integral, carry out these steps.

a. Sketch the original region of integration R in the​ xy-plane and the new region S in the​ uv-plane using the given change of variables.
b. Find the limits of integration for the new integral with respect to u and v.
c. Compute the Jacobian.
d. Change variables and evaluate the new integral.

Modifying Below Integral from nothing to nothing Integral from nothing to nothing With Upper R∫∫Rx squared StartRoot x plus 2y EndRoot 2x+2y ​dA, where Upper R equals StartSet (x comma y ): 0 less than or equals x less than or equals 2 comma negative StartFraction x Over 2 EndFraction less than or equals y less than or equals 1 minus x EndSetR=(x,y): 0≤x≤2, − x 2≤y≤1−x​; use x equals 2 ux=2u​, y equals v minus uy=v−u.

Answer 1

Answer:

The integral part of your question is not comprehensive enough. The answer provided below is for the integral;

$\int\limitsa_R \int\limitsa {x^{2} \sqrt{(x^{2}+2y)} dA$

Where:

R={(xy) 0≤x≤2, − x 2≤y≤1−x}

Explanation:

See the attached file for the calculations.