Estimate the area under the graph of f, the x-axis, and the lines x = −1 and x = 2 using three and six rectangles and a right middle left endpoints.

Question

Let f(x) = 1 + 2x²

find

r3

r6

find

l3

l6

find

m3

m6

Answer 1

f(x) = 1+2x

integrate

F(x) = x +x^2 + c

evaluate from -1 to 2

F(2)-F(-1) = 2+4 -[-1+(-1)^2]= 6-(0) = 6 = exact area

but that counts between the graph and x-axis as negative if below the x-axis

so, 6 1/2 if you count as positive

but the way the problem is worded, 6 1/4 might be what you want, the area below the graph and above the x-axis and ignore the area below the x axis and above the graph

but 6 might be what the problems wants after all. Just ignore the tricky ambiguous nuances

left and right estimations should be a less than 6 and more than 6, and closer to 6 for 6 rectangles

less so for 3 rectangles

the middle estimates should be closest, and the 6 rectangles using middle points is the closest to area of 6

graph the line and use geometry of rectangle and triangle areas

quadrant I areas sum to 6, quadrant II and III sum to 1/2 quadrant II = 1/4, quadrant III=-1/4

6+1/4-1/4=6

this tends to create some ambiguity in what the problem is asking

but use the left estimates to get 1+2(-1) + 1+2(0) + 1+2(1) = 3 for 3 rectangles

for the right estimate 3+ 1+2(2) = 9

with midpoint estimate =6

r3 =9, l3 =3, m3=6

for 6 rectangles

recntangles' bases =3/6= 1/2 instead of 1

add up the f(x)'s, the rectangles' heights and divide by 2

midpoint estimate = (1+2(-1/2) + 1+ 2(1/2) +1+ 2(3/2)/2 = 6

left estimate = (1+2(-1) + 1+2(-1/2) + 1+2(0)+ 1+ 2(1/2)+ 1+ 2(1) + 1+2(3/2)/2 = 4.5

right estimate = 15/2 = 7.5

r6 = 7.5, l6 =4.5, m6 = 6