How to 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 endpoints, left endpoints and middle enpoints

Question

Let f(x) = 3 + 2x²

find

r3

r6

find

l3

l6

find

m3

m6

Answer 1

Answer 2

1st rectangle's area with right side base= 3+ 2(0)^2= 3

2nd right side base = 3+ 2(1)^2 = 5

3rd right side base= 3+2(2)^2 = 11

total area of the 3 rectangles = 19 = r3

1st rectangle's area with left side base = 3+2(-1)^2 = 5

2nd left side base = 3+2(0)^2 = 3

3rd left side base = 3+2(1)^2 =5

total area = 13 = l3

1st rectangle's area with middle of base

= 3+2(-.5)^2 = 3+.5 = 3.5

2nd middle = 3+2(.5)^2 = 3.5

3rd middle = 3+2(1.5)^2 = 3+4.5 = 7.5

total area = 14.5 = m3

for r6, l6, and m6 use 6 rectangles each with base = 0.5

actual area = integral of 3+2x^2 = 3x+ (2/3)x^3

evaluated from x=-1 to x=2

= 3(2)+(2/3)(8) -[3(-1)+ (2/3)(-1)]

= 6+16/3 +3 +2/3

=9+ 18/3

= 9+6

= 15

using midpoints gives the closest approximation, just off by 1/2

right sides are off by 4

left off by 2

it helps to sketch the graph and points

each rectangle has a base =1

heights vary for each rectangle

y=3+2x^2 is an upward opening parabola with vertex = minimum point = (0,3)

using 6 rectangles

and middle of each base

rectangle's areas are each of the following divided by 2

3+2(-3/4)^2 =3+9/8 = 4.125

3+2(-1/4)^2 = 3+1/8 = 3.125

3+2(1/4)^2 = 3.125

3+2(3/4)^2 = 4.125

3+2(5/4)^2 =3+25/8 = 6.125

3+2(7/4)^2 =3+49/8 =9.125

total area = 29.75/2= 14.875= m6, which is closest approximation to 15, the actual exact area

areas for right end points with 6 rectangles are the following, each divided by 2

3+2(-1/2)^2 =3.5

3+ 2(0)= 3

3+2(1/2)= 3.5

3+2(1) = 5

3+2(3/2)=7.5

3 +2(4) = 11

total areas = 33.5/2 = 16.7.5= r6, off by 1.5

areas for left end points with 6 rectangles:

replace 11 with 0

total areas = 22.5/2 = 11.25 = l6, off by 3.25

the more rectangles the closer the approximations