Question

Sketch and describe the solid that is produced when the region enclosed by x = 0, y = −x, and y = −3 is rotated around the x-axis. Then _______ nd the volume of the solid.

Answer 1

The rotating region forms a conical solid, and the volume can be calculated using the formula for the volume of a cone, resulting in 9π cubic units.

Step-by-step explanation:

To sketch and describe the solid produced by rotating the region enclosed by x = 0, y = −x, and y = − 3 around the x-axis, we first visualize the delineated section on the xy-plane. This region is a triangle with vertices (0,0), (0,−3), and (3,−3). When this region is rotated around the x-axis, it produces a conical solid.To find the volume of the cone we use the formula V = 1/3πr²h, where 'r' is the radius of the base and 'h' is the height. In this case, the radius r is 3, and the height h is also 3. Plugging in these values, we compute the volume as V = 1/3π(3)²(3) = 9π cubic units.