Question

Find the volume of the solid that is bounded on the front and back by the planes x= 1 and x=5, on the sides by the cylinders y = £ and above and below by the planes z = x +4 and z=0. The volume of the bounded solid is?

Answer 1

To find the volume of the solid bounded by the given planes and cylinders, calculate the volume of the cylinder and rectangular prism separately, then subtract the volume of the cylinder from the volume of the prism.

Step-by-step explanation:

To find the volume of the solid bounded by the given planes and cylinders, we will break it down into smaller parts and calculate their volumes separately.

Step 1: Calculate the volume of the cylinder using the formula V = πr²h, where r is the radius of the cylinder and h is the height. In this case, the radius is £ and the height is the difference between z=0 and z=x+4.

Step 2: Calculate the volume of the rectangular prism between the planes x=1 and x=5. The base of this prism is the cross-sectional area of the cylinder, and the height is the difference between z=0 and z=x+4.

Step 3: Subtract the volume of the cylinder from the volume of the rectangular prism to find the volume of the bounded solid.

Remember to use the correct units for your calculations.