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.
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?
To find the volume of the solid, we need to determine its base and the shape of its cross sections perpendicular to the y-axis.
First, let's find the base of the solid. The base is bounded by the curves y = |x| + a, 0 < a < 3, and the line y = 3. To determine the limits of integration, we need to find the x-values at which the curves intersect:
1. Set y = |x| + a equal to y = 3:
|x| + a = 3
2. Solve this equation for x by considering two cases:
a) x is positive:
x + a = 3
x = 3 - a
b) x is negative:
-x + a = 3
x = a - 3
So, the x-values where the curves intersect are x = 3 - a and x = a - 3.
To determine the limits of integration for the base, we need to find the x-value where the curves intersect at y = 3. Since y = |x| + a is symmetric about the y-axis, we can take the positive x-value.
The limits of integration for the base are therefore from x = 0 to x = 3 - a.
Next, let's examine the cross sections perpendicular to the y-axis. We know that these cross sections are equilateral triangles.
Consider a small strip at y = y_0. The width of this strip is dx, and its height is given by y = |x| + a.
To find the area of the cross section at height y_0, let's determine the width of the strip using its height:
Since the cross section is an equilateral triangle, its height is the same as its side length. Thus, y_0 is also the side length of the triangle.
Using the equation y = |x| + a, we can write the equation for the triangle's side length as follows:
y_0 = |x| + a
To solve for x in terms of y_0, consider two cases:
1. x is positive:
x + a = y_0
x = y_0 - a
2. x is negative:
-x + a = y_0
x = a - y_0
So, the width of the strip at y = y_0 is given by the difference between the x-values corresponding to the positive and negative cases, which is (y_0 - a) - (a - y_0) = 2(y_0 - a).
Now, we can find the area of the cross section at height y_0 by multiplying the width and height of the strip:
Area = 1/2 * base * height
= 1/2 * (2(y_0 - a)) * y_0
= (y_0 - a) * y_0
Finally, to find the volume of the solid, we need to integrate the area function over the limits of integration for the base:
Volume = ∫[0 to 3 - a] (y_0 - a) * y_0 dx
By integrating this expression, you should be able to find the volume of the solid in terms of 'a'.