Consider the region trapped between the graphs of y=9-x^2 and y=11-3x.
a) find the area of this region
b) set up an integral which computes the volume of the solid of rotation obtained by rotating this region about the x-axis
c) set up an integral which computes the volume of the solid of rotation obtained by rotating this region about the y-axis
To solve this problem, we need to find the points of intersection between the two curves and calculate the area of the region trapped between them.
a) To find the points of intersection, we set the two equations equal to each other:
9 - x^2 = 11 - 3x
Rearranging the equation gives:
x^2 - 3x + 2 = 0
This equation can be factored as:
(x - 1)(x - 2) = 0
Setting each factor equal to zero, we find two solutions:
x - 1 = 0 --> x = 1
x - 2 = 0 --> x = 2
Next, we substitute these x-values into either equation to find the corresponding y-values.
Substituting x = 1 into y = 9 - x^2 gives:
y = 9 - (1)^2 = 9 - 1 = 8
Substituting x = 2 into y = 11 - 3x gives:
y = 11 - 3(2) = 11 - 6 = 5
Now we have the points of intersection: (1, 8) and (2, 5).
To find the area of the region, we integrate the difference between the upper curve (y = 9 - x^2) and the lower curve (y = 11 - 3x) from x = 1 to x = 2:
Area = ∫[1, 2] (11 - 3x) - (9 - x^2) dx
b) To set up an integral for the volume of the solid of rotation obtained by rotating the region about the x-axis, we use the disk method. The volume can be calculated as the integral of the cross-sectional area of each disk:
Volume = ∫[1, 2] π[(9 - x^2)^2 - (11 - 3x)^2] dx
c) To set up an integral for the volume of the solid of rotation obtained by rotating the region about the y-axis, we use the washer method. The volume can be calculated as the integral of the cross-sectional area of each washer:
Volume = ∫[5, 8] π[(r_outer)^2 - (r_inner)^2] dy
Where r_outer is the distance from the x-axis to the curve y = 9 - x^2, and r_inner is the distance from the x-axis to the curve y = 11 - 3x.
Please note that to compute the actual volumes in parts (b) and (c), you would need to evaluate the integrals.