Refer to BC CALC VOLUMES OF REVOLUTIONS:

Question

Refer to BC CALC VOLUMES OF REVOLUTIONS: Given f(x)=x^2-3 g(x)=4x+9: The region between f(x) and g(x) is the base of a solid. Find the volume of the solid if cross-sections perpendicular to the x-axis and equilateral triangles

Answer 1

Draw the base of the solid. It is bounded above by the line y = 4x + 9 and below by the parabola y = x²-3, The curves intersect at x = -2 and x = 6,

The area of an equilateral triangle is (√3/4)s², where s is the length of a side

In this problem, s = (4x+9) - (x²-3) = -x²+4x+12

The volume is found by integrating (√3/4)s² from x = -2 to x = 6.