Final answer:
To sketch the region bounded by the curves y=x^2 and y=3, we plot the two curves on a coordinate system and find the shaded area between them. To set up the integral needed to find the volume of the solid generated by revolving this region about the x-axis using the shell method, we slice the region into infinitesimally thin cylindrical shells and set up the integral as V = 2π ∫[a,b] x * (3 - x^2) dx.
Step-by-step explanation:
To sketch the region bounded by the curves y=x^2 and y=3, we first plot the two curves on a coordinate system. The curve y=x^2 is a parabola that opens upward, and the curve y=3 is a horizontal line at y=3. The region bounded by these curves is the shaded area between the two curves.
To set up the integral needed to find the volume of the solid generated by revolving this region about the x-axis using the shell method, we imagine slicing the region into infinitesimally thin cylindrical shells. The radius of each shell is equal to the x-coordinate of a point on the curve y=x^2, and the height of each shell is the difference between the y-coordinates of the two curves at that x-coordinate.
Therefore, the integral needed to find the volume of the solid can be set up as follows: V = 2π ∫[a,b] x * (3 - x^2) dx, where [a,b] are the x-values of the intersections of the two curves.