Final answer:
To find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis, we can use the method of discs or washers.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis, we can use the method of discs or washers.
First, let's find the points of intersection between the curves y = 4 - (3/2)x and y = 0. Setting the two equations equal to each other, we have 4 - (3/2)x = 0. Solving for x, we get x = 8/3. Since the curves intersect at x = 8/3 and the region is bounded by the lines x = 0 and x = 1, this is the range of integration for our problem.
Now we can set up the integral to find the volume. We will integrate the area of cross-sections perpendicular to the x-axis from x = 0 to x = 8/3. The cross-sections will be circles with radii given by the y-values of the curves. The radius can be obtained by solving each equation for y and taking the difference: r = (4 - (3/2)x) - 0 = 4 - (3/2)x.
Using the formula for the volume of a cylinder, V = πr^2h, where h is the height of the cylinder, we can express the volume of one cross-section as V = π(4 - (3/2)x)^2*dx. Integrating this expression from x = 0 to x = 8/3 will give us the total volume.