Areas of regions Find the area of the following regions. x - x?(x + 1)(x² + 1)The region bounded by the curve yand thex-axis from x = 0 to x = 1

Answered: x - x? (x + 1)(x² + 1) The region…

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x - x? (x + 1)(x² + 1) The region bounded by the curve y and the x-axis from x = 0 to x = 1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.1: Polynomial Functions Of Degree Greater Than

Problem 12E

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Question

Areas of regions Find the area of the following regions.

x - x?
(x + 1)(x² + 1)
The region bounded by the curve y
and the
x-axis from x = 0 to x = 1

Expert Solution

Step 1

Area of the region bounded by f, the x axis and the vertical lines x = a and x = b is,

$A = \int_{a}^{b} f (x) d x$

Step 2

Given that,

$y = f (x) = \frac{x - x^{2}}{(x + 1) (x^{2} + 1)}$

Step 3

Firstly simplify $\frac{x - x^{2}}{(x + 1) (x^{2} + 1)}$ using partial fraction method, so

$\frac{x - x^{2}}{(x + 1) (x^{2} + 1)} = \frac{A}{x + 1} + \frac{B x + C}{x^{2} + 1} . . . (1) x - x^{2} = A (x^{2} + 1) + (B x + C) (x + 1) P u t x = - 1, - 1 - 1 = A (2) \Rightarrow - 2 = 2 A \Rightarrow A = - 1 P u t x = 0, 0 = A + C \Rightarrow C = - A \Rightarrow C = 1 P u t x = 1, 1 - 1 = A (2) + (B + C) (2) \Rightarrow A + B + C = 0 \Rightarrow B = - A - C \Rightarrow B = - 1 - 1 \Rightarrow B = - 2$

Substitute the value of A, B and C in (1),

$\frac{x - x^{2}}{(x + 1) (x^{2} + 1)} = \frac{- 1}{x + 1} + \frac{- 2 x + 1}{x^{2} + 1} \frac{x - x^{2}}{(x + 1) (x^{2} + 1)} = - \frac{1}{x + 1} - \frac{2 x}{x^{2} + 1} + \frac{1}{x^{2} + 1} . . . (2)$

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