Final answer:
To find the region bounded by the curves x+y=2, 4x+y=8, and x=0, we need to determine the points of intersection between the curves. These points are (2, 0), (0, 2), and (0, 8). The volume of the solid generated by revolving this region about the y-axis can be found using the method of cylindrical shells, and is equal to 16π.
Step-by-step explanation:
Solution
To find the region bounded by the curves, we need to determine the points of intersection between the curves.
- Find the points of intersection between x+y=2 and 4x+y=8:
Subtracting the first equation from the second equation, we get: 3x = 6. Therefore, x = 2. Plugging this value into either equation gives us y = 0. So, the point of intersection is (2, 0). - Find the points of intersection between x+y=2 and x=0:
Plugging x=0 into the equation x+y=2, we get y=2. So, the point of intersection is (0, 2). - Find the points of intersection between 4x+y=8 and x=0:
Plugging x=0 into the equation 4x+y=8, we get y=8. So, the point of intersection is (0, 8).
Now we can sketch the region bounded by the curves:
To find the volume of the solid generated by revolving this region about the y-axis, we can use the method of cylindrical shells. The volume of each shell is given by 2πrh, where r is the distance from the y-axis to the shell, and h is the height of the shell. Since the region is symmetric about the y-axis, we can consider only the region in the first quadrant. The height of each shell is given by the difference between the top and bottom curves, which is (4x+y)-(x+y)=3x. The distance from the y-axis to the shell is x. Therefore, the volume of each shell is 2πx(3x) = 6πx^2.
To find the total volume, we need to integrate the volume of each shell over the range of x from 0 to 2 (the points of intersection). Therefore, the volume of the solid generated by revolving this region about the y-axis is given by the integral of 6πx^2 from 0 to 2:
∫(6πx^2)dx, where the limits of integration are 0 and 2.
Integrating 6πx^2 with respect to x gives us 2πx^3. Evaluating this expression at the limits of integration gives us (2π(2^3) - 2π(0^3)) = 16π.
Therefore, the volume of the solid generated by revolving this region about the y-axis is 16π.