(1,1)(2,1)(2,3)
mirror around y-axis & revolve
solve with Shell and washer
please and thank you
revolve around what? And why reflect it as well?
I will say that the line from (1,1) to (2,3) is
y = 2x-1
You can use that when doing the volume of a shell, which is v = 2πrh dx
revolve around y
well, you are still vague. I assume you mean to revolve the triangle around the y-axis.
So, revolve the small triangle. The volume is
v = ∫[1,2] 2πrh dx
where r = x and h = y-1 = 2x-2
v = ∫[1,2] 2πx(2x-2) dx = 10π/3
to verify, do it again with discs of thickness dy, where
v = ∫[1,3] π(R^2-r^2) dy
where R=2 and r=x=(y+1)/2
v = ∫[1,3] π(2^2-((y+1)/2)^2) dy = 10π/3
To mirror the given points around the y-axis, we need to flip the x-coordinate of each point while keeping the y-coordinate unchanged.
(1,1) becomes (-1,1)
(2,1) becomes (-2,1)
(2,3) becomes (-2,3)
To revolve these mirrored points around the y-axis, we can imagine rotating the resulting curve formed by these points around the y-axis.
To solve this using the shell and washer method, we need to find the volume of the solid formed by revolving the curve around the y-axis.
The shell method involves integrating the circumference of cylindrical shells formed by slicing the solid parallel to the y-axis. The washer method involves integrating the difference between the outer and inner radii of cylindrical washers formed by slicing the solid parallel to the y-axis.
For this particular problem, it's important to note that the curve formed by the mirrored points is not a function, hence we cannot solve it using a single integral. We need to split it into two parts: one for the first half and one for the second half.
First, let's consider the region from x = -2 to x = 0 (the left half of the curve). To solve using the shell method, we would integrate:
∫[x1, x2] 2πx f(x) dx,
where x1 = -2 and x2 = 0.
For the shell method, the distance between a shell and the y-axis is x, and the height of each shell is given by f(x). So our integral becomes:
∫[-2,0] 2πx (3-x) dx.
To solve using the washer method, we would calculate the volume of each washer as the difference between the outer radius (x) and the inner radius (f(x)). The integral becomes:
∫[-2,0] π(x^2 - (3-x)^2) dx.
Next, let's consider the region from x = 0 to x = 2 (the right half of the curve). We would use the same process as above, but with the limits reversed.
For the shell method, the integral becomes:
∫[0,2] 2πx (x-1) dx.
For the washer method, the integral becomes:
∫[0,2] π(x^2 - (x-1)^2) dx.
Finally, we add the volumes of the left and right halves to get the total volume of the solid formed by revolving the curve around the y-axis.
I hope this helps! Let me know if you have any further questions.