Final answer:
To sketch the region bounded by the curves y=6x√, x+y=7, and y=3, find the points of intersection and plot them on a graph. The region will be a rectangle. To find the volume of the solid generated by revolving this region about the x-axis, use the method of cylindrical shells and integrate over the interval [0.5, 1].
Step-by-step explanation:
To sketch the region bounded by the curves y=6x√, x+y=7, and y=3, start by finding the points of intersection of these curves. Set y=6x√ equal to y=3 and solve for x to find x=0.5. Next, set y=6x√ equal to x+y=7 and solve for x to find x=1. Plug these x-values back into the equations to find the corresponding y-values. Now, plot the points on a graph and sketch the curves accordingly. The region bounded by these curves will be a rectangle.
To find the volume of the solid generated by revolving this region about the x-axis, we can use the method of cylindrical shells. The radius of each cylindrical shell will be the distance from the x-axis to the curve y=6x√. The height of each cylindrical shell will be the difference between the y-value of the curve x+y=7 and the y-value of the curve y=3. So, the volume of each cylindrical shell is given by V=2πrh, where r is the radius and h is the height. Integrate this expression over the interval [0.5, 1] to find the total volume of the solid.