Final answer:
To sketch the region bounded by the curves x y=5,y=0 and x=0, plot the hyperbola x*y=5, the x-axis y=0, and the y-axis x=0. To find the volume using the shell method, consider cylindrical shells with thickness dx, radius x, and height 5-x. Integrate the volume of each shell from x=0 to x=5.
Step-by-step explanation:
To sketch the region bounded by the curves x y=5,y=0 and x=0, we begin by graphing the three curves on a coordinate plane. The curve x*y=5 is a hyperbola that opens upwards, with asymptotes y=0 and x=0. The curve y=0 is the x-axis, and the curve x=0 is the y-axis. These three curves intersect at the point (5, 1).
To find the volume of the solid generated by revolving this region about the y-axis using the shell method, we need to consider thin cylindrical shells with thickness dx. Each shell has a radius equal to the distance from the y-axis, which is x. The height of each shell is the difference between the values of y on the curves x*y=5 and y=0, which is 5-x. The volume of each shell is given by V = 2πx(5-x)dx.
To find the total volume, we integrate the volume of each shell from x=0 to x=5. The integral is: V = ∫[0 to 5] (2πx(5-x))dx.