Question

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

Answer 1

v = ∫2πrh dx
where r=x and h=y
v = ∫[78/7,90/7] 2πx(6/5 - 7/5 |x-12|) dx = 8634π/35

Answer 2

This is wrong, thank you

Answer 3

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

Answer 4

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.

Answer 5

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.