(a) Estimate the area under the graph of f ( x ) = 3 + 2 x 2 from x = 1 to x = 2
ID: 2889739 • Letter: #
Question
(a) Estimate the area under the graph of
f(x) = 3 + 2x2 from x = 1 to x = 2
using three rectangles and right endpoints.
R3 =
Then improve your estimate by using six rectangles.
R6 =
Sketch the curve and the approximating rectangles for R3.
Sketch the curve and the approximating rectangles for R6.
(b) Repeat part (a) using left endpoints.
Sketch the curve and the approximating rectangles for L3.
Sketch the curve and the approximating rectangles for L6.
(c) Repeat part (a) using midpoints.
Sketch the curve and the approximating rectangles for M3.
Sketch the curve and the approximating rectangles for M6.
(d) From your sketches in parts (a)-(c), which appears to be the best estimate?
M6L6 R6
Explanation / Answer
f = 3 + 2x^2
x = -1 to x = 2
a = -1 abd b = 2
So, deltax = (b - a)/n
n = 3 :
deltax = 3/3 = 1
So, endpoints are -1 , 0 , 1 and 2
L3 :
Left endpts are -1 , 0 and 1
Adding function values :
3 + 2(-1)^2 + 3 + 2(0)^2 + 3 + 2(1)^2
13
To this multiply deltax :
13 * 1
13
R3 :
Right endpts are 0 , 1 and 2
Adding fn values :
3 + 2(0)^2 + 3 + 2(1)^2 + 3 + 2(2)^2
19
To this multiply deltax :
19 * 1
19
M3 :
Endpts are -1 , 0 , 1 and 2
Midpoints are -0.5 , 0.5 and 1.5
Adding function values :
3 + 2(-0.5)^2 + 3 + 2(0.5)^2 + 3 + 2(1.5)^2
14.5
To this mu;tiply deltax :
14.5 * 1
14.5
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L6 :
delta(x) = 0.5
Endpts are -1 , -0.5 , 0 , 0.5 , 1 , 1.5 and 2
L6 :
Adding elft end function values :
3 + 2(-1)^2 + 3 + 2(-0.5)^2 + 3 + 2(0)^2 + 3 + 2(0.5)^2 + 3 + 2(1)^2 + 3 + 2(1.5)^2
27.5
To this multiply deltax :
27.5 * 0.5
13.75
R6 :
Adding right end values :
3 + 2(-0.5)^2 + 3 + 2(0)^2 + 3 + 2(0.5)^2 + 3 + 2(1)^2 + 3 + 2(1.5)^2 + 3 + 2(2)^2
33.5
To this multiply deltax :
33.5 * 0.5
16.75
M6 :
Midpts are : -0.75 , -0.25 , 0.25 , 0.75 , 1.25 and 1.75
Adding these function values :
3 + 2(-0.75)^2 + 3 + 2(-0.25)^2 + 3 + 2(0.25)^2 + 3 + 2(0.75)^2 + 3 + 2(1.25)^2 + 3 + 2(1.75)^2
29.75
To this multiply deltax :
29.75 * 0.5
to get 14.875
M6 appears most reliable
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