Use cylindrical shells to find the volume formed by rotating the region in the first quadrant enclosed by: y=1.2-1.4|x-12| and y=0 about the y-axis
v = ∫2πrh dx
where r=x and h=y
v = ∫[78/7,90/7] 2πx(6/5 - 7/5 |x-12|) dx = 8634π/35
This is wrong, thank you
Well, clearly you did not bother to evaluate the integral yourself. I made a typo, and the answer is 864π/35 instead.
Just to check, we can apply the theorem of Pappus. The area of the triangular region is (1/2)(6/5)(12/7) = 36/35
The centroid is at x=12, so the distance traveled during the revolution is 24π
24π * 36/35 = 864π/35
Ok. Thank you. You are right but I tried working it out using 2π∫(11.142-12.857) (x(18-1.4x) dx.
I got a different answer, Is there a way I am to set up the y-equation.
well, you know the slope on one side is 1.4 and the slope on the other side is -1.4
You have the point (12,1.2) so the region has two lines for its boundary:
y = 1.4(x-12)+1.2 = -15.6+1.4x
y = -1.4(x-12)+1.2 = 18-1.4x
Type in the functions at wolframalpha.com and you can see that the 1.2-1.4|x-12| looks quite different from your region.
To find the volume formed by rotating the region in the first quadrant enclosed by the curves y = 1.2 - 1.4|x - 12| and y = 0 about the y-axis using cylindrical shells, you can follow these steps:
1. Find the limits of integration: Determine the x-values at which the two curves intersect. Set them equal to each other and solve for x.
1.2 - 1.4|x - 12| = 0
|x - 12| = 1.2 / 1.4
x - 12 = ±0.857
x = 12 ± 0.857
x = 12.857, 11.143
So, the limits of integration will be from x = 11.143 to x = 12.857.
2. Set up the integral: The volume element, or the width of each cylindrical shell, is given by Δx. The radius of each shell will be the distance from the y-axis to the function y = 1.2 - 1.4|x - 12|.
The volume of each cylindrical shell can be expressed as V = 2πh(x)r(x)Δx, where h(x) is the height of the shell and r(x) is the radius.
In this case, h(x) will be the difference between the top function (y = 1.2 - 1.4|x - 12|) and the bottom function (y = 0), so h(x) = 1.2 - 1.4|x - 12| - 0 = 1.2 - 1.4|x - 12|.
The radius r(x) will be the distance from the y-axis (x = 0) to the function y = 1.2 - 1.4|x - 12|. Since the curve is symmetric about the y-axis, we can consider the positive side and double the result.
r(x) = x = 12.857 - x for x ≥ 12 or r(x) = -x + 12 for x < 12.
Therefore, the integral to find the volume will be:
V = ∫[11.143 to 12.857] 2π(1.2 - 1.4|x - 12|)(12.857 - x) dx + ∫[12 to ∞] 2π(1.2 - 1.4|x - 12|)(-x + 12) dx
3. Evaluate the integral: Perform the integration for each interval separately.
For the first interval [11.143 to 12.857]:
V1 = ∫[11.143 to 12.857] 2π(1.2 - 1.4|x - 12|)(12.857 - x) dx
For the second interval [12 to ∞]:
V2 = ∫[12 to ∞] 2π(1.2 - 1.4|x - 12|)(-x + 12) dx
Finally, the total volume will be V = V1 + V2.
Depending on your preferred method of evaluating definite integrals, you can either solve these integrals analytically or use numerical methods.