MATH 1014 Tutorial Notes 8

Transcription

1 MATH4 Calculus II (8 Spring) Topics covered in tutorial 8:. Numerical integration. Approximation integration What you need to know: Midpoint rule & its error Trapezoid rule & its error Simpson s rule & its error Midpoint rule: MATH 4 Tutorial Notes 8 Suppose x is defined and integrable on [, ], the Midpoint rule approximation to xx using n equally spaced subintervals on [, ] is M = [ ( + ) + ( + ) + ( + )] where x =, x n k = + x for =,, n. Error bounds for Midpoint Rule: Assume that is continuous on the interval [, ], then the error in approximating the integral xx by Midpoint Rule with n subintervals satisfy the inequality: K E M, hr K r Example 8. Use Midpoint Rule with n = to approximate the integral x x Page of 3

2 MATH4 Calculus II (8 Spring) Trapezoid rule: Suppose x is defined and integrable on [, ], the Trapezoid rule approximation to xx using n equally spaced subintervals on [, ] is = [ ] where x =, x n k = + x for =,, n. Error bounds for Trapezoid Rule: Assume that is continuous on the interval [, ], then the error in approximating the integral xx by Trapezoid Rule with n subintervals satisfy the inequality: K E, hr K r Example 8. Consider the integral approximation of x 4x. a) Does overestimate or underestimate the exact value? b) Find the error bound for without calculating. 8 Example 8.3 (Volume on Numerical integration) The region bounded by the graph of the function x = n x and the x-axis over the interval [, ] is rotated about x-axis so that the volume of the solid of revolution obtained can be represented by an integral of the form Fxx. If the trapezoid rule on four subintervals of equal length is used to estimate this integral, which of the following values is the approximate volume of the solid thus found? Page of 3

3 MATH4 Calculus II (8 Spring) Simpson s rule: Suppose x is defined and integrable on [, ], the Simpson s rule approximation to xx using n equally spaced subintervals on [, ] is = [ ] where n is an even integer, and x =, x k = + x for =,, n. n Error bounds for Simpson s Rule: Assume that 4 is continuous on the interval [, ], then the error in approximating the integral xx by Sipso s Rule with n subintervals satisfy the inequality: K E 8, hr K r Example 8.4 Given the following definite integral and n =, answer the following questions. (Round your answers to six decimal places.) sin x x a) Use the Midpoint Rule to approximate the definite integral. M 4 =? b) Use the Trapezoid Rule to approximate the definite integral. 4 =? c) Use Sipso s Rule to approximate the definite integral. 4 =? Page 3of 3

4 Extra exercises Midpoint Rule: b a Trapezoidal Rule: Zeta CHAN: Numerical Integration f ( x) dx midpoint Riemann sum b b a f ( x) dx [ f ( x ) f ( x ) f ( x ) f ( xn ) a n f ( x n )]. Let y = p (x) = ax + bx + c intersects y = f (x) at ( h, y ), (, y ) and (h, y ). (a) Express the coefficients a, b and c in terms of h, y, y and y. (b) Express the net area under y = p (x) on the interval [ h, h] in terms of h, y, y and y. (c) Write the approximation for b a f ( x) dx of the Simpson s rule.. Consider the definite integral cos xdx. (a) Find its exact value. (b) Find its approximation using the midpoint rule with 6 subintervals. (c) Find its approximation using the trapezoidal rule with 6 subintervals. (d) Find its approximation using the Simpson s rule with 6 subintervals. 3. Estimate the following definite integrals using the midpoint rule, the trapezoidal rule and the Simpson s rule with the indicated number of subintervals. (a) x dx with 4 subintervals (b) 9 x 3 dx with 4 subintervals (c) sin xdx with 6 subintervals (d) e x dx with 8 subintervals b a b a Midpoint Rule: absolute error max f ( t) 4 n atb b a b a Trapezoidal Rule: absolute error max f ( t) n atb b a b a (4) Simpson s Rule: absolute error max f ( t) 8 n atb 4 Page of

5 Extra exercises Zeta CHAN: 4. For each of the following definite integrals with the indicated number of subintervals, find an upper bound for the absolute error if (i) the midpoint rule, (ii) the trapezoidal rule or (iii) the Simpson s rule is used to estimate it. Correct to 4 decimal places if necessary. (a) x dx with 4 subintervals (b) 9 x 3 dx with 4 subintervals (c) sin xdx with 6 subintervals (d) e x dx with 8 subintervals 5. Show that the Simpson s rule approximation to the definite integral of any cubic function on any closed interval is exact. e dx, at least how many subintervals are needed so that the absolute error is less than.5 if each of the following rules is used? (a) the midpoint rule (b) the trapezoidal rule (c) the Simpson s rule 6. To estimate the entry P( Z ) in the standard normal distribution table, i.e. x Page of