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 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

.