Question

how do i find the finite approximation to estimate the area using the lower sum of 4 rectangles for f(x)= 4-x^2 between x=-2 and x=2?

Answer 1

well i see an easy way to make 4 rectangles from -2 to 2. they each have a width of 1.

Answer 2

and they have a height of the minimum between f(x) and f(x+1), ie the left and right sides

Answer 3

i do not see a way to obtain a lower sum
some of the rectangles end up above the curve starting from left endpoint or starting from right endpoint

Answer 4

i guess a lower sum does not mean all the rectangles have to be under the curve

Answer 5

well the first rectangle would be from -2 to -1.
f(-2) = 0
f(-1) = 2

this rectangle has 0 height, therefore it has no area. A = length * width = 0.

second rectangle from -1 to 0
f(-1) = 2
f(0) = 4

this rectangle has height of 2 (the minimum). area = height * width = 2 * 1 = 2

third rectangle from 0 to 1 is the same
A3 = 2

fourth rectangle from 1 to 2 is the same as the first:
A4 = 0 * 1 = 0

Total area = A1 + A2 + A3 + A4 = 4

Answer 6

starting from right endpoint we do
Ar=1*(f(2)+f(1)+f(0)+f(-1)]
starting from left end point we do
Al=1*(f(-2)+f(-1)+f(0)+f(1)]

Answer 7

but these Ar=Al since f is even function

Answer 8

sweet, thankyou both

Answer 9

oh wait Heisenberg i see
i was only thinking of left endpoint rule and right endpoint rule
and i even though of the mid point rule
but we we could have chosen any number to approximate the area
and since we want a lower sum we look in each of the intervals (-2,-1), (-1,0), (0,1), (1,2)
and pick the sample number that gives the minimum height
great job Heisenberg

Answer 10

ah yes. i didn't even think about those. luckily for me, it worked out. ...this time :p