Final answer:
To find the area of the surface that lies within the cylinder x^2+y^2=9, set up a double integral over the region defined by the cylinder and evaluate it.
Step-by-step explanation:
To find the area of the surface that lies within the cylinder, we need to evaluate the double integral over the region defined by the cylinder. The surface is given by the equation z = 3x^2 + 3y^2 and the cylinder is given by the equation x^2 + y^2 = 9.
Step 1: Set up the limits of integration for x and y. Since the cylinder is defined by x^2 + y^2 = 9, we can express y as a function of x: y = sqrt(9 - x^2). The limits of integration for x are -3 to 3, which correspond to the boundaries of the cylinder.
Step 2: Set up the double integral: ∬D (1 + (dz/dx)^2 + (dz/dy)^2) dA, where D is the region defined by the cylinder and dA is the differential area element.
Step 3: Evaluate the double integral over the region D using the limits of integration and the equation for z.
The calculated double integral will give you the area of the surface that lies within the cylinder.