To evaluate the given double integral over the region 𝐷D, bounded by the lines 𝑦=−𝑥y=−x, 𝑦=𝑥2y=x 2 , and 𝑦=2y=2,

To evaluate the given double integral over the region D, bounded by the lines y=-x, y=x^2, and y=2, follow these steps:

1. Identify the region D: This is the area in the xy-plane bounded by the lines y=-x, y=x^2, and y=2. Sketching these lines will help you visualize the region.

2. Set up the limits of integration: The lower limit is the curve y=x^2 and the upper limit is the line y=2. For the x-integration, the lower limit is the line y=-x and the upper limit is the line y=x.

3. Write down the double integral: The double integral over the region D is ∫∫D f(x,y) dA, where f(x,y) is the function you're integrating (not given in the question), and dA is the differential area element.

4. Evaluate the inner integral: This is the integral with respect to y, from y=x^2 to y=2.

5. Evaluate the outer integral: This is the integral with respect to x, from x=-y to x=y.

6. Compute the final result: After evaluating the inner and outer integrals, you'll have the value of the double integral over the region D.

Note: Without the specific function to integrate, I can't provide a more detailed solution.