7.7 Approximate Integration 1-6. Find midpoint, trapezoidal Rule, Simpson’s Rule.

1. , . (a) 0.235977  (b)0.232771  (c)0.233760

2. , . (a) 2.612462  (b) 2.576704  (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 do we have to choose so that the approximations and to the integral in part (a) are accurate to within 0.00001? (a) , .

(b) .

(c) .

For , we choose so that solving this, so that .

For , we 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 do we have to choose so that the approximations , , and . to the integral in part (a) are accurate to within 0.00001? (a)  , , .

(b) Since (3) and (4) 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      .