Answer:
To find the points of intersection between the two equations, we set them equal to each other:
3x² = 12
Dividing both sides by 3, we get:
x² = 4
Taking the square root of both sides, we obtain:
x = ±2
So the points of intersection are (-2, 12) and (2, 12).
To graph the region bounded by y = 3x² and y = 12, we plot the points of intersection and the key points for both parabolas.
For y = 3x², we can take a few x-values and find their corresponding y-values:
For x = -1, y = 3(-1)² = 3(1) = 3 For x = 0, y = 3(0)² = 3(0) = 0 For x = 1, y = 3*(1)² = 3(1) = 3
The key points for the parabola y = 3x² are (-1, 3), (0, 0), and (1, 3).
For y = 12, the y-value is always 12 for any x-value. Therefore, the key point for the line y = 12 is (0, 12).
Now we can graph the region:
y-axis |
| . | . | . |-----------------------
| . | . | . |------------------------------------- | ∙ (0, 12) | |-2---1---0----1----2----3----4---x-axis
The parabola y = 3x² will be concave upward and will intersect the line y = 12 symmetrically at points (-2, 12) and (2, 12). The region bounded by y = 3x² and y = 12 is the shaded region in the graph.
Now let's use the disc/washer method to set up the integral that computes the volume of the solid formed by revolving this region about the line y = 12.
To use the disc/washer method, we will consider infinitely thin discs or washers (annular regions) and sum up their volumes.
Since we are revolving the region about the line y = 12, we can see that the radius of each disc or washer will be the distance from the line y = 12 to the curve y = 3x².
The curve y = 3x² is below y = 12 for the region we are interested in. Therefore, the radius will be 12 - y = 12 - 3x².
The height of each disc or washer will be the differential of the x-axis, which is dx.
The area of each disc or washer will be π(radius)² = π(12 - 3x²)².
To calculate the volume of the solid, we integrate the area of each disc/washer from the leftmost point of intersection (-2) to the rightmost point of intersection (2):
V = ∫[from -2 to 2] π(12 - 3x²)² dx
Step-by-step explanation: