the exact value of a definite integral is 5. the following questions refer to estimates of this integral...?

Question

...using the left, right, trapezoid and simpson's rules. use what you know about approximate error to answer the following questions.

suppose left(40)=2.3055 estimate left(160).
suppose trap(40)=2.7250 estimate trap(160)
suppose simp(40)=3.1680 estimate simp(160)


HELP!
i'm very much confused.
explain it step by step, please.
i am really trying to learn this stuff.

Answer 1

This question is about understanding how the error terms for these rules vary with the step size. We are going from 40 points to 160 points, so the step size will be 1/4 of the original value.

Left and right Riemann sums have an error term proportional to the step size. Since the original error is 2.3055 - 5 = -2.6945, the new error should be about -2.6945/4 = -0.6734, so the result should be about 5 - 0.6734 = 4.3266.

The trapezoid rule has an error proportional to the square of the step size. So the new error should be 1/16 of the original. The original error is 2.7250 - 5 = -2.2750, so the new error should be about -2.2750/16 = -0.1422, so the result should be about 5 - 0.1422 = 4.8578.

Simpson's rule has an error proportional to the fourth power of the step size. The original error is -1.8320, so the new error should be about -1.8320/64 = -0.0286. So the new estimate should be about 4.9714.

Answer 2

i'm sorry yet Gianlino's info isn't valid. x^2 and sqrt(x) are inverse of one yet another, even with the shown fact that their integrals from 0 to a million are respectively a million/3 and a couple of/3 greater frequently, the graphs of applications inverse of one yet another would be symmetrical wrt the 1st bisector. as a result the integrals, measuring the section between those graphs and the x axis and the strains x=0 and x=a million, will frequently not be equivalent. they are able to be so basically particularly situations. i think of that's extremely undemanding. enable me have a attempt. Dina, your substitution is very very nearly the mind-blowing one, seem With u^3 = a million-x^7 J = Int(0,a million)(a million-x^7)^(a million/3) dx transforms as follows: (a million-x^7)^(a million/3) --> u dx --> d(a million-u^3)^(a million/7) So J = Int(a million,0)ud(a million-u^3)^(a million/7) = u(a million-u^3)^(a million/7)|_1^0 - Int_1^0 (a million-u^3)^(a million/7)du = 0 + your 2d essential so as that they are equivalent ! you should use the comparable trick for any (m,n) advantageous integers in decision to (3,7) Edited: ok, my argument approximately gianlino's replaced into incorrect through fact i did not pay interest to his specifying reducing applications. And the substitute of variables I proposed is strictly changing function and variable (subsequently making use of the inverse function) and is an illustration of gianlino's declare. i choose some cafeine