Final answer:
The question involves finding the volume of a solid in the first octant bounded by a cylinder and a plane using double integration. The volume is found by setting up a double integral over the x-y domain, taking into account that z varies between 0 and 3-y, and x and y are bounded by the cylinder.
Step-by-step explanation:
The student has asked to use double integration to find the volume of the solid bounded by a cylinder, a plane, and the coordinate planes in the first octant. This problem involves setting up and evaluating a double integral, accounting for the equations that describe each boundary. In this case, we have the cylinder x²+y²=4, the plane z+y=3, and since the solid is in the first octant, z and y must be non-negative.
First, we establish the bounds for y and x by noting the radius of the cylinder in the x-y plane which is 2 (since x²+y²=4 gives us a radius r of √4 or 2). In the first octant, x and y must also be non-negative. Then we observe the relationship between y and z that is given by z+y=3, meaning z varies from 0 to 3-y.
Our double integral becomes:
∫0∫0√(4-y²)3-y dz dx dy
To solve this, we integrate with respect to z first, then with respect to x, and finally with respect to y, resulting in the volume of the solid.