47–52. Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 2 47. (2x + 1) dx 48. (1 — х) dx 7 49. (4x + 6) dx 50. (x2 – 1) dx 3 .4 .2 51. (x² – 1) dx 52. 4x3 dx

47–52. Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 2 47. (2x + 1) dx 48. (1 — х) dx 7 49. (4x + 6) dx 50. (x2 – 1) dx 3 .4 .2 51. (x² – 1) dx 52. 4x3 dx

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.4: Values Of The Trigonometric Functions

Problem 22E

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Question

Q51, calculate the definite integrals using the limit of both right and left Riemann sums. Calculate the left Riemann sums two (2) ways.

47-52. Limits of sums Use the definition of the definite integral to
evaluate the following definite integrals. Use right Riemann sums and
Theorem 5.1.
.2
47.
(2x + 1) dx
48.
(1 — х) dx
7
(4x + 6) dx
(x2 – 1) dx
49.
50.
3
.4
51.
(x² – 1) dx
52.
4x3 dx

THEOREM 5.1 Sums of Powers of Integers
Let n be a positive integer and c a real number.
п(п + 1)
n
п
Ec =
Ek
с — сп
2
k=1
k=1
п(n + 1)(2n + 1)
n²(n + 1)²
п
Σ2
6.
4
k=1
k=1

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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