Asked by Adriana
Find the volume of the solid whose base is the region bounded by the x-axis, the curves y=x, y=3x^2, x=0, and x=.333333 and which has the property that each cross section perpendicular to the x-axis is an equilateral triangle.
Answers
Answered by
Steve
You could have just said the region bounded by the two curves, as they intersect at the two given x values.
Each cross-section has a base equal to the distance between the curves, or x-3x^2
Since the area of an equilateral triangle of side s is s^2√3/4, you just have to integrate to add 'em all up to make a volume:
v = ∫[0,1/3] s^2√3/4 dx
where s = x-3x^2
v = ∫[0,1/3] √3/4 (x-3x^2)^2 dx = 1/(1080√3)
Each cross-section has a base equal to the distance between the curves, or x-3x^2
Since the area of an equilateral triangle of side s is s^2√3/4, you just have to integrate to add 'em all up to make a volume:
v = ∫[0,1/3] s^2√3/4 dx
where s = x-3x^2
v = ∫[0,1/3] √3/4 (x-3x^2)^2 dx = 1/(1080√3)
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.