Fundamental Theorem

Question

Use the Fundamental Theorem of Calculus to find the "area under curve" of y= -x^2+9x between x=3 and x=6. In you calculations , if you need to round, do not do so until the very end of the problem.

Answer 1

First, draw the graph of y = -x² + 9x. It is a parabola opening downward with x-intercepts (0,0) and (9,0). The highest point is (4.5, 20.25). Between x = 3 and x = 6, the graph lies above the x-axis.

Area = ∫_{(from 3 to 6)} [-x² + 9x]dx = [-(1/3)x³ + 4.5x²]_{(from 3 to 6)}

= [(-1/3)(216) + 4.5(36)] - [(-1/3)(27) + 4.5(9)] = -(1/3)(216 - 27) + 4.5(36 - 9) = 58.5