Explanation When area is enclosed by two curves y = f ( x ) and y = g ( x ) area is calculated using vertical elements by subtracting the lower curve from the upper curve and evaluating the integral. Given: The curves y = 1 1 + x 2 ----------- (i) y = x 2 ------ (ii) Trace the curves on X Y plane with scale X Maximum 4 X Minimum -4 X Scale 1 Y Maximum 3 Y Minimum -3 Y Scale 1 The region OABCO is the area of enclosed between the curves (i) & (ii) Points of intersection of the curves (i) & (ii) Substitute (ii) in (i) y = 1 1 + y Solve for y y ( 1 + y ) = 1 y + y 2 = 1 y + y 2 − 1 = 0 We know that roots of quadratic a y 2 + b y + c = 0 are − b ± b 2 − 4 a c 2 a Here a = 1 , b = 1 and c = − 1 y = − 1 ± ( 1 ) 2 − 4 ( 1 ) ( − 1 ) 2 ( 1 ) y = − 1 − 5 2 = − 3...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

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Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

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Question

Chapter 6.1, Problem 55E

To determine

The integral that represents the area between the curves $y = \frac{1}{1 + x^{2}}$ and $y = x^{2}$

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Chapter 6.1, Problem 54E

Chapter 6.1, Problem 56E

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Chapter 6 Solutions

CALCULUS (CLOTH)

Ch. 6.1 - Prob. 1PQ Ch. 6.1 - Prob. 2PQ Ch. 6.1 - Prob. 3PQ Ch. 6.1 - Prob. 4PQ Ch. 6.1 - Prob. 5PQ Ch. 6.1 - Prob. 6PQ Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E

Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - Prob. 13E Ch. 6.1 - Prob. 14E Ch. 6.1 - Prob. 15E Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Prob. 53E Ch. 6.1 - Prob. 54E Ch. 6.1 - Prob. 55E Ch. 6.1 - Prob. 56E Ch. 6.1 - Prob. 57E Ch. 6.1 - Prob. 58E Ch. 6.1 - Prob. 59E Ch. 6.1 - Prob. 60E Ch. 6.1 - Prob. 61E Ch. 6.1 - Prob. 62E Ch. 6.1 - Prob. 63E Ch. 6.1 - Prob. 64E Ch. 6.2 - Prob. 1PQ Ch. 6.2 - Prob. 2PQ Ch. 6.2 - Prob. 3PQ Ch. 6.2 - Prob. 4PQ Ch. 6.2 - Prob. 5PQ Ch. 6.2 - Prob. 1E Ch. 6.2 - Prob. 2E Ch. 6.2 - Prob. 3E Ch. 6.2 - Prob. 4E Ch. 6.2 - Prob. 5E Ch. 6.2 - Prob. 6E Ch. 6.2 - Prob. 7E Ch. 6.2 - Prob. 8E Ch. 6.2 - Prob. 9E Ch. 6.2 - Prob. 10E Ch. 6.2 - Prob. 11E Ch. 6.2 - Prob. 12E Ch. 6.2 - Prob. 13E Ch. 6.2 - Prob. 14E Ch. 6.2 - Prob. 15E Ch. 6.2 - Prob. 16E Ch. 6.2 - Prob. 17E Ch. 6.2 - Prob. 18E Ch. 6.2 - Prob. 19E Ch. 6.2 - Prob. 20E Ch. 6.2 - Prob. 21E Ch. 6.2 - Prob. 22E Ch. 6.2 - Prob. 23E Ch. 6.2 - Prob. 24E Ch. 6.2 - Prob. 25E Ch. 6.2 - Prob. 26E Ch. 6.2 - Prob. 27E Ch. 6.2 - Prob. 28E Ch. 6.2 - Prob. 29E Ch. 6.2 - Prob. 30E Ch. 6.2 - Prob. 31E Ch. 6.2 - Prob. 32E Ch. 6.2 - Prob. 33E Ch. 6.2 - Prob. 34E Ch. 6.2 - Prob. 35E Ch. 6.2 - Prob. 36E Ch. 6.2 - Prob. 37E Ch. 6.2 - Prob. 38E Ch. 6.2 - Prob. 39E Ch. 6.2 - Prob. 40E Ch. 6.2 - Prob. 41E Ch. 6.2 - Prob. 42E Ch. 6.2 - Prob. 43E Ch. 6.2 - Prob. 44E Ch. 6.2 - Prob. 45E Ch. 6.2 - Prob. 46E Ch. 6.2 - Prob. 47E Ch. 6.2 - Prob. 48E Ch. 6.2 - Prob. 49E Ch. 6.2 - Prob. 50E Ch. 6.2 - Prob. 51E Ch. 6.2 - Prob. 52E Ch. 6.2 - Prob. 53E Ch. 6.2 - Prob. 54E Ch. 6.2 - Prob. 55E Ch. 6.2 - Prob. 56E Ch. 6.2 - Prob. 57E Ch. 6.2 - Prob. 58E Ch. 6.2 - Prob. 59E Ch. 6.2 - Prob. 60E Ch. 6.2 - Prob. 61E Ch. 6.2 - Prob. 62E Ch. 6.2 - Prob. 63E Ch. 6.2 - Prob. 64E Ch. 6.2 - Prob. 65E Ch. 6.2 - Prob. 66E Ch. 6.2 - Prob. 67E Ch. 6.2 - Prob. 68E Ch. 6.3 - Prob. 1PQ Ch. 6.3 - Prob. 2PQ Ch. 6.3 - Prob. 3PQ Ch. 6.3 - Prob. 4PQ Ch. 6.3 - Prob. 1E Ch. 6.3 - Prob. 2E Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 4E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 6E Ch. 6.3 - Prob. 7E Ch. 6.3 - Prob. 8E Ch. 6.3 - Prob. 9E Ch. 6.3 - Prob. 10E Ch. 6.3 - Prob. 11E Ch. 6.3 - Prob. 12E Ch. 6.3 - Prob. 13E Ch. 6.3 - Prob. 14E Ch. 6.3 - Prob. 15E Ch. 6.3 - Prob. 16E Ch. 6.3 - Prob. 17E Ch. 6.3 - Prob. 18E Ch. 6.3 - Prob. 19E Ch. 6.3 - Prob. 20E Ch. 6.3 - Prob. 21E Ch. 6.3 - Prob. 22E Ch. 6.3 - Prob. 23E Ch. 6.3 - Prob. 24E Ch. 6.3 - Prob. 25E Ch. 6.3 - Prob. 26E Ch. 6.3 - Prob. 27E Ch. 6.3 - Prob. 28E Ch. 6.3 - Prob. 29E Ch. 6.3 - Prob. 30E Ch. 6.3 - Prob. 31E Ch. 6.3 - Prob. 32E Ch. 6.3 - Prob. 33E Ch. 6.3 - Prob. 34E Ch. 6.3 - Prob. 35E Ch. 6.3 - Prob. 36E Ch. 6.3 - Prob. 37E Ch. 6.3 - Prob. 38E Ch. 6.3 - Prob. 39E Ch. 6.3 - Prob. 40E Ch. 6.3 - Prob. 41E Ch. 6.3 - Prob. 42E Ch. 6.3 - Prob. 43E Ch. 6.3 - Prob. 44E Ch. 6.3 - Prob. 45E Ch. 6.3 - Prob. 46E Ch. 6.3 - Prob. 47E Ch. 6.3 - Prob. 48E Ch. 6.3 - Prob. 49E Ch. 6.3 - Prob. 50E Ch. 6.3 - Prob. 51E Ch. 6.3 - Prob. 52E Ch. 6.3 - Prob. 53E Ch. 6.3 - Prob. 54E Ch. 6.3 - Prob. 55E Ch. 6.3 - Prob. 56E Ch. 6.3 - Prob. 57E Ch. 6.3 - Prob. 58E Ch. 6.3 - Prob. 59E Ch. 6.3 - Prob. 60E Ch. 6.3 - Prob. 61E Ch. 6.3 - Prob. 62E Ch. 6.3 - Prob. 63E Ch. 6.3 - Prob. 64E Ch. 6.3 - Prob. 65E Ch. 6.3 - Prob. 66E Ch. 6.4 - Prob. 1PQ Ch. 6.4 - Prob. 2PQ Ch. 6.4 - Prob. 3PQ Ch. 6.4 - Prob. 1E Ch. 6.4 - Prob. 2E Ch. 6.4 - Prob. 3E Ch. 6.4 - Prob. 4E Ch. 6.4 - Prob. 5E Ch. 6.4 - Prob. 6E Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 8E Ch. 6.4 - Prob. 9E Ch. 6.4 - Prob. 10E Ch. 6.4 - Prob. 11E Ch. 6.4 - Prob. 12E Ch. 6.4 - Prob. 13E Ch. 6.4 - Prob. 14E Ch. 6.4 - Prob. 15E Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 19E Ch. 6.4 - Prob. 20E Ch. 6.4 - Prob. 21E Ch. 6.4 - Prob. 22E Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - Prob. 25E Ch. 6.4 - Prob. 26E Ch. 6.4 - Prob. 27E Ch. 6.4 - Prob. 28E Ch. 6.4 - Prob. 29E Ch. 6.4 - Prob. 30E Ch. 6.4 - Prob. 31E Ch. 6.4 - Prob. 32E Ch. 6.4 - Prob. 33E Ch. 6.4 - Prob. 34E Ch. 6.4 - Prob. 35E Ch. 6.4 - Prob. 36E Ch. 6.4 - Prob. 37E Ch. 6.4 - Prob. 38E Ch. 6.4 - Prob. 39E Ch. 6.4 - Prob. 40E Ch. 6.4 - Prob. 41E Ch. 6.4 - Prob. 42E Ch. 6.4 - Prob. 43E Ch. 6.4 - Prob. 44E Ch. 6.4 - Prob. 45E Ch. 6.4 - Prob. 46E Ch. 6.4 - Prob. 47E Ch. 6.4 - Prob. 48E Ch. 6.4 - Prob. 49E Ch. 6.4 - Prob. 50E Ch. 6.4 - Prob. 51E Ch. 6.4 - Prob. 52E Ch. 6.4 - Prob. 53E Ch. 6.4 - Prob. 54E Ch. 6.4 - Prob. 55E Ch. 6.4 - Prob. 56E Ch. 6.4 - Prob. 57E Ch. 6.4 - Prob. 58E Ch. 6.4 - Prob. 59E Ch. 6.4 - Prob. 60E Ch. 6.4 - Prob. 61E Ch. 6.4 - Prob. 62E Ch. 6.4 - Prob. 63E Ch. 6.4 - Prob. 64E Ch. 6.5 - Prob. 1PQ Ch. 6.5 - Prob. 2PQ Ch. 6.5 - Prob. 3PQ Ch. 6.5 - Prob. 4PQ Ch. 6.5 - Prob. 1E Ch. 6.5 - Prob. 2E Ch. 6.5 - Prob. 3E Ch. 6.5 - Prob. 4E Ch. 6.5 - Prob. 5E Ch. 6.5 - Prob. 6E Ch. 6.5 - Prob. 7E Ch. 6.5 - Prob. 8E Ch. 6.5 - Prob. 9E Ch. 6.5 - Prob. 10E Ch. 6.5 - Prob. 11E Ch. 6.5 - Prob. 12E Ch. 6.5 - Prob. 13E Ch. 6.5 - Prob. 14E Ch. 6.5 - Prob. 15E Ch. 6.5 - Prob. 16E Ch. 6.5 - Prob. 17E Ch. 6.5 - Prob. 18E Ch. 6.5 - Prob. 19E Ch. 6.5 - Prob. 20E Ch. 6.5 - Prob. 21E Ch. 6.5 - Prob. 22E Ch. 6.5 - Prob. 23E Ch. 6.5 - Prob. 24E Ch. 6.5 - Prob. 25E Ch. 6.5 - Prob. 26E Ch. 6.5 - Prob. 27E Ch. 6.5 - Prob. 28E Ch. 6.5 - Prob. 29E Ch. 6.5 - Prob. 30E Ch. 6.5 - Prob. 31E Ch. 6.5 - Prob. 32E Ch. 6.5 - Prob. 33E Ch. 6.5 - Prob. 34E Ch. 6.5 - Prob. 35E Ch. 6.5 - Prob. 36E Ch. 6.5 - Prob. 37E Ch. 6.5 - Prob. 38E Ch. 6.5 - Prob. 39E Ch. 6.5 - Prob. 40E Ch. 6.5 - Prob. 41E Ch. 6.5 - Prob. 42E Ch. 6.5 - Prob. 43E Ch. 6 - Prob. 1CRE Ch. 6 - Prob. 2CRE Ch. 6 - Prob. 3CRE Ch. 6 - Prob. 4CRE Ch. 6 - Prob. 5CRE Ch. 6 - Prob. 6CRE Ch. 6 - Prob. 7CRE Ch. 6 - Prob. 8CRE Ch. 6 - Prob. 9CRE Ch. 6 - Prob. 10CRE Ch. 6 - Prob. 11CRE Ch. 6 - Prob. 12CRE Ch. 6 - Prob. 13CRE Ch. 6 - Prob. 14CRE Ch. 6 - Prob. 15CRE Ch. 6 - Prob. 16CRE Ch. 6 - Prob. 17CRE Ch. 6 - Prob. 18CRE Ch. 6 - Prob. 19CRE Ch. 6 - Prob. 20CRE Ch. 6 - Prob. 21CRE Ch. 6 - Prob. 22CRE Ch. 6 - Prob. 23CRE Ch. 6 - Prob. 24CRE Ch. 6 - Prob. 25CRE Ch. 6 - Prob. 26CRE Ch. 6 - Prob. 27CRE Ch. 6 - Prob. 28CRE Ch. 6 - Prob. 29CRE Ch. 6 - Prob. 30CRE Ch. 6 - Prob. 31CRE Ch. 6 - Prob. 32CRE Ch. 6 - Prob. 33CRE Ch. 6 - Prob. 34CRE Ch. 6 - Prob. 35CRE Ch. 6 - Prob. 36CRE Ch. 6 - Prob. 37CRE Ch. 6 - Prob. 38CRE Ch. 6 - Prob. 39CRE Ch. 6 - Prob. 40CRE Ch. 6 - Prob. 41CRE Ch. 6 - Prob. 42CRE Ch. 6 - Prob. 43CRE Ch. 6 - Prob. 44CRE Ch. 6 - Prob. 45CRE Ch. 6 - Prob. 46CRE Ch. 6 - Prob. 47CRE Ch. 6 - Prob. 48CRE Ch. 6 - Prob. 49CRE Ch. 6 - Prob. 50CRE Ch. 6 - Prob. 51CRE Ch. 6 - Prob. 52CRE Ch. 6 - Prob. 53CRE Ch. 6 - Prob. 54CRE

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