1549901369-Elements_of_Real_Analysis__Denlinger_

7.2 The Riemann Integral Defined 371 Repeat Exercise 6 for the function f(x) = x^2 + 3x. Prove Theorem 7.2. 12 (b). Prove Theor ...

372 Chapter 7 • The Riemann Integral Prove that Theorem 7.2.14 remains true if "< c" is replaced by ":S c". Prove Theorem 7. ...

7.3 The Integral as a Limit of Riemann Sums 373 Proof. Suppose f is defined and bounded on [a, b]. Then 3M > 0 3 \:/x E [a, b ...

374 Chapter 7 • The Riemann Integral Adding together inequalities (3) and (4), we have E S(f, Q) - S.(f, Q) < S(f, P) - S. ...

7.3 The Integral as a Limit of Riemann Sums 375 Lemma 7.3.4 For any partition P of [a, b], and any selection of tags xi, x;_, · ...

376 Chapter 7 11 The Riemann Integral Part 2 ( {:::: ): Suppose that Ve > 0, 3 8 > 0 3 V tagged partitions P* of [a, b], l ...

7.3 The Integral as a Limit of Riemann Sums 377 THE INTEGRAL AS A LIMIT Note: Because of Theorem 7.3.5 it makes sense to write n ...

378 Chapter 7 • The Riemann Integral Example 7.3.7 Use the technique of Theorem 7.3.6 to calculate J 1 4 (x^2 - 4x + 5)dx. Solut ...

7.3 The Integral as a Limit of Riemann Sums 379 Definition 7.3.8 Suppose f is defined and bounded on [a, b], where a < b. For ...

380 Chapter 7 • The Riemann Integral The next theorem and its consequences provide conclusive evidence that regular partitions a ...

7.3 The Integral as a Limit of Riemann Sums 381 x t Yk 1 - It Yk 1 Yk 1 +1 f krlt Yk 2 Yk 2 +1 Yk,-2 x, Yk 2 -2 x2 Figure 7.6 Pu ...

382 Chapter 7 • The Riemann Integral Proof. (a) By definition, l: f =sup mu, P) : Pis a partition of [a, b]}. Let c: > 0. By ...

7.3 The Integral as a Limit of Riemann Sums 383 Theorem 7.3.15 (Regular Partition Riemann/Darboux Criterion for Integrability) A ...

384 Chapter 7 • The Riemann Integral [ Xi-l, xi] created by Q are equal (to 1). Further, all trapezoidal^6 approxima- tions and ...

7.3 The Integral as a Limit of Riemann Sums 385 11. Modify the techniques used in Exercise 10, as needed, to evaluate each of th ...

386 Chapter 7 111 The Riemann Integral 7 .4 Basic Existence and Additivity Theorems Learning the results of this section may be ...

7.4 Basic Existence and Additivity Theorems 387 Theorem 7.4.2 (Additivity of the Integral, I) If f is integrable on [a, b] then ...

388 Chapter 7 • The Riemann Integral Proof. Suppose f is integrable on [a, c] and on [c, b], where a < c < b. Then f is bo ...

7.4 Basic Existence and Additivity Theorems 389 . Since f is integrable on [a+t:, b-t:], the middle terms of the right-hand side ...

390 Chapter 7 • The Riemann Integral The next theorem is really rather remarkable, and perhaps unexpected. The result shows the ...