Calculus-Sec-7-7-Solution
7.7 Approximate Integration and CAS by SGLee - HSKim, JYLee
1-6. Find the integral using the midpoint, trapezoidal and Simpson’s rule for the given 
.
1. 
, 
Use sin(x^3) in http://matrix.skku.ac.kr/cal-lab/Area-Sum.html with midpoint and trapezoidal rules.
(a) 0.232771 with midpoint rule (b) 0.2359771 with trapezoidal rule (c) 0.233760
2. 
, 
. (
2.58862863250718)
Use e^(3*x)*sin(2*x) in http://matrix.skku.ac.kr/cal-lab/Area-Sum.html with midpoint and trapezoidal rules.
(a) 2.57670 with midpoint rule (b) 2.612462 with trapezoidal rule <Error = 0.0238343387259241>
(c) 2.588559
3. 
, 
(a) 0.919952 (b) 0.927027 (c) 0.925237
4. 
, 
(a) 0.272198 (b) 0.272198 (c) 0.272198
5. 
, 
(a) 0.457277 (b) 0.458528 (c) 0.458114
6. 
, 
(a) 1.182973 (b) 1.160116 (c) 1.169130
7. (a) Determine the approximations 
and 
for 
.
(b) Find the errors involved in the approximations of part (a).
(c) Determine how large must 
be so that the approximations 
and 
to the integral in part (a) are accurate to within 0.00001?
(a) 
, 
.
(b) 
.
(c)
.
For 
, we must choose 
so that 
solving this, 
so that 
.
For 
, we must choose 
so that 
solving this gives, 
so that 
.
8.(a) Determine the approximations and for 
and 
for 
and the corresponding errors 
and 
.
(b) Compare the actual errors in part (a) with the error estimates given by and
(c) Determine how large must we choose 
so that the approximations 
, 
, and 
for the integral in part (a) are accurate to within 0.00001?
(a) 
, 
, 
, 
.
(b) Since 
and gives
,
.
(c) For 
, find 
so that 
, 
.
For 
, find 
so that 
, 
.
For 
, find 
so that 
, 
.
9. Given the function 
at the following values,
|
|
1.8 |
1.9 |
2.0 |
2.1 |
2.2 |
2.3 |
2.4 |
|
|
0.028561 |
0.020813 |
0.015384 |
0.011525 |
0.008742 |
0.006709 |
0.004079 |
approximate 
using Simpson's Rule.
Plot 
and use Simpson's Rule
.
