Consider the function #f(x)= -(x-3)^2+4# how do you write an equation using a limit to determine the area enclosed by f(x) and the x-axis?

1 Answer
Aug 14, 2017

#A = lim_(nrarroo)sum_(i=1)^n (-(i4/n-2)^2+4) 4/n#

Explanation:

The curve intersects the #x# axis at #f(x) =0#

or #x=1# and #x=5#.

Cut the interval #[1,5]# into #n# pieces each of length #(5-1)/n = 4/n#

The right endpoints of the subintervals are #1+(4i)/n#.

We can approximate the area under the curve on the #i^(th)# interval using a rectangle of

base #4/n# and

height #f(1+(4i)/n) = -(1+(4i)/n-3)^2+4 = -(i4/n-2)^2+4#.

We then sum the areas of the rectangles #sum_(i=1)^n (-(i4/n-2)^2+4) 4/n#.

Finally, we take a limit as the subintervals get shorter and shorter (go to #0#). This is aso the limit as the number of rectangles increases without bound (#nrarroo#)

#A = lim_(nrarroo)sum_(i=1)^n (-(i4/n-2)^2+4) 4/n#