(a) Use the Midpoint Rule with n = 10 to approximate the integral. (b) Give an upper bound for the error involved in this approximation.
(a) Use the Midpoint Rule with n = 10 to approximate the integral. (b) Give an upper bound for the error involved in this approximation.
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![EXAMPLE 3
(a) Use the Midpoint Rule with n = 10 to approximate the integral.
(b) Give an upper bound for the error involved in this approximation.
SOLUTION
(a) Since a = 0, b = 2, and n = 10, the Midpoint Rule gives
* Ax[f(0.1) + f( 0.3
v ) + ... + f( 1.7
v ) + f(1.9)]
+ e0.81
e3.61]
0.2[e0.01 + e
0.09
+ e0.25
e2.25
0.49
=
+ e
+ el.21
* 16.097
+ e
1.69
2.89
+
+ e
+
Video Example )
The figure illustrates this approximation.
(b) Since (x) =
ex2, we have '(x) =
2
2xe
v and "(x) = 2|2x?+1]-e*
'et?
v . Also, since 0 < x < 2 we have
x s 4
v and so
0 sf "(x) = | 18e
X < 18e4
Taking K = 18e“, a = 0, b = 2, and n = 10 in the error estimate, we see that an
upper bound for the error is as follows. (Round your answer to five decimal
places.)
18e* · 8
- 3.2758 V
(10)2
24](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb13ad392-c0f2-4a1f-8c28-c5aad8f5d77d%2Fa13ca64e-abbc-47d5-9cfe-9a8703c163a4%2F4aqc0wb_processed.png&w=3840&q=75)
Transcribed Image Text:EXAMPLE 3
(a) Use the Midpoint Rule with n = 10 to approximate the integral.
(b) Give an upper bound for the error involved in this approximation.
SOLUTION
(a) Since a = 0, b = 2, and n = 10, the Midpoint Rule gives
* Ax[f(0.1) + f( 0.3
v ) + ... + f( 1.7
v ) + f(1.9)]
+ e0.81
e3.61]
0.2[e0.01 + e
0.09
+ e0.25
e2.25
0.49
=
+ e
+ el.21
* 16.097
+ e
1.69
2.89
+
+ e
+
Video Example )
The figure illustrates this approximation.
(b) Since (x) =
ex2, we have '(x) =
2
2xe
v and "(x) = 2|2x?+1]-e*
'et?
v . Also, since 0 < x < 2 we have
x s 4
v and so
0 sf "(x) = | 18e
X < 18e4
Taking K = 18e“, a = 0, b = 2, and n = 10 in the error estimate, we see that an
upper bound for the error is as follows. (Round your answer to five decimal
places.)
18e* · 8
- 3.2758 V
(10)2
24
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