Show that S 2 also gives the exact value for b a x 3 dx and conclude, as in Exercise 63, that S N is exact for all cubic polynomials Show by counterexample that S 2 is not exact for integrals of x 4 Data From Exercise 63 For N even, divide a, b into N subintervals of width x b a N Set x j a j x, y ...

Data From Exercise 62 get Let fx x Then Ax b a2 and the endpoints are ...

Show that S 2 also gives the exact value for b a x 3 dx and

Question:

Show that S₂also gives the exact value for ∫^b_ax³ dx and conclude, as in Exercise 63, that S_Nis exact for all cubic polynomials. Show by counterexample that S₂ is not exact for integrals of x⁴.

Data From Exercise 63

For N even, divide [a, b] into N subintervals of width Δx = b − a/N. Set x_j= a + j Δx, y_j= ƒ(xj), and

s2= b-a 3N (V2j + 4y2j+1 + y2j+2)

(a) Show that Sy is the sum of the approximations on the intervals [X2, X2j+2]that is, Sy = S + S2 + ... +

Data From Exercise 62

that Show that if f(x) = px + qx +r is a quadratic polynomial, then S = = 6 f(x where yo = = f(a), y = f a +

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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Question Posted: Jan 15, 2024 07:00 AM

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Show that S 2 also gives the exact value for b a x 3 dx and

Question:

s2= b-a 3N (V2j + 4y2j+1 + y2j+2)

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Data From Exercise 62 get Let fx x Then Ax b a2 and the endpoints are ...View the full answer

Calculus

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