Find the volume of the solid generated by revolving the region about the given line.
The region in the second quadrant bounded above by the curve y = 16 - x2, below by the x-axis, and on the right by the y-axis, about the line x = 1
I have gathered, that washer method is to be used - (-4,0) is the shaded area.
washers is a good way:
v = ∫[0,16] π(R^2-r^2) dy
where R=1+|x|, r=1
v = π∫[0,16] (1+√(16-y))^2 - 1 dy
= -π(4/3 (16-y)^(3/2) + 1/2 (16-y)^2) [0,16]
= 640π/3
Or, using shells,
v = ∫[-4,0] 2πrh dx
where r=1-x and h=y
v = 2π∫[-4,0] (x+1)(16-x^2) dx
= 2πx (x^3/4 - x^2/3 - 8x + 16)[-4,0]
= 640π/3
Hmm! shells is less complicated this time.
∫(1+√(16-y))^2 - 1 dy
let u = 16-y
du = -dy
∫(1+√u)^2 - 1 du
∫1 + 2√u + u - 1
∫2√u + u
2(2/3)u^(3/2) + 1/2 u^2
because
∫u^n = 1/(n+1) u^(n+1) where n=1/2
Where does the 4/3 and 3/2 come from?
Thank you
To find the volume of the solid generated by revolving the given region about the line x = 1, we can indeed use the washer method. The region in the second quadrant bounded above by the curve y = 16 - x^2, below by the x-axis, and on the right by the y-axis, should be revolved about the line x = 1.
First, let's visualize the region and the solid generated. The region can be described as a quarter of a circle with radius 4 (since y = 16 - x^2 and x is restricted to be non-negative in the second quadrant). When this region is revolved about the line x = 1, it generates a solid that is an annular disk or a "washer."
To find the volume of each "washer," we need to consider the difference between the outer and inner radii. The outer radius, r_outer, is the distance from the line x = 1 to the curve y = 16 - x^2. The inner radius, r_inner, is the distance from the line x = 1 to the x-axis.
To calculate the outer radius:
We know that x = 1, and y = 16 - x^2. Substituting x = 1 into the equation gives us y = 16 - 1^2 = 15. Therefore, the outer radius is 15 units.
To calculate the inner radius:
We can visualize that the inner radius is the distance from the line x = 1 to the x-axis, which is simply 1 unit.
Now, we can set up the integral for the volume using the washer method. To calculate the volume, we need to integrate the difference between the outer and inner areas over the range of x-values that describe the region.
Using the washer method, the formula for the volume is:
V = π ∫[a,b] (r_outer^2 - r_inner^2) dx,
where [a, b] represents the range of x-values that describe the region.
Since we have already determined that the region is from x = 0 to x = 4 (as given by the coordinates (-4, 0) and (0, 0)), we can integrate the difference in areas from x = 0 to x = 4.
Thus, the volume can be calculated as:
V = π ∫[0,4] [(15^2) - (1^2)] dx.
Evaluating this integral will give you the volume of the solid generated by revolving the given region about the line x = 1.