Have mercy!
Find the volume of the solid obtained by rotating y=3+2x-x^2, x+y=3 about the y-axis.
better integrate on x, since doing discs on y will need to separate the region into two pieces. So, using shells,
v = ∫[0,3] 2πrh dx
where r = x and h = (3+2x-x^2) - (3-x)
v = 2π∫[0,3] x((3+2x-x^2) - (3-x)) dx
= 2π∫[0,3] 3x^2-x^3 dx
= 2π(x^3 - x^4/4) [0,3]
= 27/2 π
Thank you!!!
If you would check my last 2 questions. I am just completely stumped.
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To find the volume of the solid obtained by rotating a curve around the y-axis, you can use the method of cylindrical shells.
First, let's graph the two equations y = 3 + 2x - x^2 and x + y = 3 to visualize the region we need to rotate.
To find the points of intersection, we can set the two equations equal to each other:
3 + 2x - x^2 = 3 - x
2x - x^2 + x = 0
x(2 - x) + x = 0
(2 - x)(x + 1) = 0
From this, we find two solutions: x = 2 and x = -1.
Now, let's calculate the volume of the solid using cylindrical shells.
The general formula for the volume of a solid using cylindrical shells is:
V = ∫(2πrh) dx
where r is the distance from the axis of rotation to the shell, h is the height of the shell, and dx represents the thickness of the shell.
In this case, the radius (r) will be the x-value of the curve, and the height (h) will be the difference between the upper and lower curves.
For the curve y = 3 + 2x - x^2, the upper curve is y = 3 + 2x - x^2, and the lower curve is y = 3 - x.
To find the volume, we need to integrate the expression 2πrh with respect to x over the interval where the curves intersect.
V = ∫[lower limit - upper limit] (2πx) (3 + 2x - x^2 - (3 - x)) dx
= ∫[lower limit - upper limit] (2πx) (3 + 3x - x^2) dx
= ∫[lower limit - upper limit] (6πx + 6πx^2 - 2πx^3) dx
= 6π ∫[lower limit - upper limit] (x + x^2 - x^3) dx
Integrating each term separately, we get:
V = 6π [ (x^2/2) + (x^3/3) - (x^4/4) ] evaluated from x = -1 to x = 2
Plugging in these values, we have:
V = 6π [ ((2^2)/2) + ((2^3)/3) - ((2^4)/4) - ((-1^2)/2) - ((-1^3)/3) + ((-1^4)/4) ]
= 6π [ 2 + 8/3 - 4 - 1/2 + 1/3 - 1/4]
= 6π [ 8/3 - 9/4]
= 6π [ (32 - 27) / 12]
= 6π [ 5 / 12]
= 5π / 2
Therefore, the volume of the solid obtained by rotating the given curves around the y-axis is (5π / 2) cubic units.