Problema Solution
Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y = 6 x^2 and y = x^2 + 9 . Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region?
Answer provided by our tutors
This is a fancy way of asking you to find the area of the region bounded by two curves x^2 + 9 and 6x^2.
x^2 + 9 is above 6x^2, so subtract 6x^2 from x^2 + 9 and integrate from -1.9 to 1.9. (from x^2 + 9 = 6x^2 we find the points x = -19 and x = 1.9)
∫ (x^2 + 9) - (6x^2) dx = ∫ -5x^2 + 9 dx = -5 ∫ x^2 dx + 9∫ dx = (- 5/3) x^3 + 9x
(- 5/3) x^3 + 9x evaluate from -1.9 to 1.9
(- 5/3) (1.9)^3 + 9*1.9 + (5/3) (-1.9)^3 - 9*(-1.9) = 11.37 is the area of the enclosed region.