Question

22 (1 point) a) The rectangles in the graph below illustrate a left endpoint Riemann sum for f(x) on the interval (2,6]. 9 The value of this Riemann sum is and this Riemann sum is an underestimate of the area of the region enclosed by y = f(x), the x-axis, and the vertical lines x = 2 and x = 6. y 8 7 6 5 4 3 2 1 X 1 2 3 4 5 6 7 8 Left endpoint Riemann sum for y = on (2, 6]

Answer 1

The left endpoint Riemman sum denotation is fiven by

Area & Lη =Σ(1-1) Δα i=1

= f(το) Δα+ f(αι)Δz + f(x2)ΔΣ + ... + f(n-1)Δε

A.c [f(x0) + f(01) + f(22) + ... + f(n-1)]

Width of each rectangle
b - α και Δ.r. η
  
.....a <<b

n - number of rectangles

  
γι + 0 = x

Note: Co = a

Given : f(0) = 9

where, ze 2,6] => 2 < C < 6

Hence a = 2 , b = 6

Here in this graph , there are 8 rectangles of equal width . Hence n =8 ,

• 
  b - α 6 – 2 και Δα, = = | = 0.5 η 8
  
• 
  γι + 0 = x
    
  Note: Co = a
  

11 = 2+1*0.5 = 2 + 0.5 = 2.5

C2 = 2 + 2 * 0.5 = 2+1=3

C3 = 2 + 3 * 0.5 = 2 +1.5 = 3.5

04 = 2+4*0.5 = 2 + 2 = 4

C5 = 2 + 5 * 0.5 = 2 + 2.5 = 4.5

6 = 2 + 6 * 0.5 = 2+ਤੇ = 5 5

17 = 2 + 7*0.5 = 2 +3.5 = 5.5

• 
  2 f(0) = 9
  

4 f(x) = f(2) 9

2.52 f(11) = f(2.5) = 6.25 9 9

f(22) = f(3) = 9

3.52 12.25 f(13) = f(3.5) = 9 9

16 f(14) = f(4) 9 9

4.52 20.25 f(15) = f(4.5) = 9 9

25 f(16) = f(5) = 9

5.52 30.25 f(27) = f(5.5) = 9 9

Since here n = 8 we proceed up to

Area Ar [f(20) + f(21) + f(22) + f(13) + f(14) + f(15) + f(16) + f(27)

4 6.25 9 12.25 16 25 Area - 0.5 + + + + 20.25 9 + 30.25 9 + 9 9 9 9 9 9

123 = 0.5 9

==0.5*13.667

Area 6.8334.sq.units
​​​​​​​

b)

The Right endpoint Riemman sum denotation is given by

Area In Σf(x;) Δη i=1

= f(αι)ΔΙ+ f(x2) Δz + ... + f(x)Δα

= Ar [f(21) + f(22) + ... + f(en)

Width of each rectangle
b - α και Δ.r. η
  
.....a <<b

n - number of rectangles

  
γι + 0 = x

• 
  b - α 6 – 2 και Δα, = = | = 0.5 η 8
  
• 
  γι + 0 = x
  

11 = 2+1*0.5 = 2 + 0.5 = 2.5

C2 = 2 + 2 * 0.5 = 2+1=3

C3 = 2 + 3 * 0.5 = 2 +1.5 = 3.5

04 = 2+4*0.5 = 2 + 2 = 4

C5 = 2 + 5 * 0.5 = 2 + 2.5 = 4.5

6 = 2 + 6 * 0.5 = 2+ਤੇ = 5 5

17 = 2 + 7*0.5 = 2 +3.5 = 5.5

Og = 2 +8 * 0.5 = 2+4= 6

• 
  2 f(0) = 9
  ​​​​​​​

2.52 f(11) = f(2.5) = 6.25 9 9

f(22) = f(3) = 9

3.52 12.25 f(13) = f(3.5) = 9 9

16 f(14) = f(4) 9 9

4.52 20.25 f(15) = f(4.5) = 9 9

25 f(16) = f(5) = 9

5.52 30.25 f(27) = f(5.5) = 9 9

  

Since here n = 8 we proceed up to

$Area=\bigtriangleup x\left [f(x_{1})+f(x_{2})+f(x_{3})+f(x_{4})+f(x_{5})+f(x_{6})+f(x_{7})+f(x_{8}) \right ]$

$=0.5\left [ \frac{155}{9}\right ]$

==0.5*17.223