Question

Find the volume of the region in the first octant bounded by the coordinate planes, the plane y+z=7, and the cylinder x = 49-y?

Answer 1

To calculate the volume of the specified region, set up a triple integral with the bounds determined by the equations of the plane and the cylinder, integrating in the order of dxdydz.

Step-by-step explanation:

To find the volume of the region in the first octant bounded by the coordinate planes, the plane y+z=7, and the cylinder x=49-y^2, we need to integrate over the appropriate bounds. Since the plane and the cylinder intersect where y+z=7 and z=7-y, we can rewrite the cylinder's equation as x=49-(7-z)^2. Setting up the integral for volume, we have:

• • Integration with respect to z is from 0 to 7-y.
  • Integration with respect to y is from 0 to 7.
  • Integration with respect to x is from 0 to 49-(7-y)^2.

The triple integral for volume is given by V = ∫∫∫ dx dy dz. Evaluate these integrals in the order of dxdydz to find the desired volume.