3. Prove properties (3) and (4) analytically, by making x = -t in the part ofthe Integral from -a to 0 and using the definitions (1) and (2). val) is said to be even iff(-x) = f(x),(1)(x) is said to be odd iff(-x) = -f(x).(2)xample, x² and cos x are even, and x and sin x are odd. The graph ofren function is symmetric about the y-axis, as shown in Figure 48, andraph of an odd function is skew-symmetric (Figure 49). By putting x = 0,we see that an odd function always has the property that f (0) = 0. It isfrom the figures that[f(x)dx = 2[f(x)dxif f(x)is even,(3)-a(x)dx = 0 if f(x)is odd,(4)-a

Answered: 3. Prove properties (3) and (4)…

3. Prove properties (3) and (4) analytically, by making x = -t in the part of the Integral from -a to 0 and using the definitions (1) and (2).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 5E: 5. Prove that the equation has no solution in an ordered integral domain.

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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 this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.

Riemann Integral

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.

Question

3. Prove properties (3) and (4) analytically, by making x = -t in the part of
the Integral from -a to 0 and using the definitions (1) and (2).

val) is said to be even if
f(-x) = f(x),
(1)
(x) is said to be odd if
f(-x) = -f(x).
(2)
xample, x² and cos x are even, and x and sin x are odd. The graph of
ren function is symmetric about the y-axis, as shown in Figure 48, and
raph of an odd function is skew-symmetric (Figure 49). By putting x = 0
,we see that an odd function always has the property that f (0) = 0. It is
from the figures that
[f(x)dx = 2[f(x)dx
if f(x)is even,
(3)
-a
(x)dx = 0 if f(x)is odd,
(4)
-a

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