Question

measured in g/cm. Evaluate all integrals by hand, showing all your work and giving an exact answer. (a) Draw a graph of the given region and include it with your written work. (b) Set up and evaluate a double integral using the most appropriate order of integration to determine the mass of the region. (c) Determine the center of mass of the region.

Answer 1

To evaluate the mass of the region and find its center of mass, we need to set up and evaluate a double integral using the given mass density function. We then use the formula to determine the coordinates of the center of mass.

Step-by-step explanation:

To evaluate the mass of the region, we need to set up and evaluate a double integral. Let's assume the mass density of the region is given by ρ(x, y). First, we draw a graph of the given region to understand its boundaries.

Next, we set up the double integral over the region. Since the mass density is given by ρ(x, y)=poxy, the double integral becomes:

∫∫ rpoxy dA

where the limits of integration are determined by the boundaries of the region. We evaluate the double integral by integrating with respect to x and y, following the appropriate order of integration.

Finally, to find the center of mass of the region, we use the formula:

(x, y) = (∫∫ x ρ(x, y) dA / ∫∫ ρ(x, y) dA, ∫∫ y ρ(x, y) dA / ∫∫ ρ(x, y) dA)