Question
The base of a solid is bounded by y =|x|+a, 0<a<3, and the line y=3. find in cu. units in terms of a, the volume of the solid if every cross section perpendicular to the y-axis is an equilateral triangle.
Answers
Steve
each cross-section has base 2x and altitude x√3
So, we want to add up all those triangles
v = ∫[a,3] 1/2 * 2x * x√3 dy
= √3 ∫[a,3] x^2 dy
But,
y = |x| + a
y-a |x|
(y-a)^2 = x^2
v = √3 ∫[a,3] (y-a)^2 dy
and now it's cake, right?
So, we want to add up all those triangles
v = ∫[a,3] 1/2 * 2x * x√3 dy
= √3 ∫[a,3] x^2 dy
But,
y = |x| + a
y-a |x|
(y-a)^2 = x^2
v = √3 ∫[a,3] (y-a)^2 dy
and now it's cake, right?