Expressing an Integral as a sum of terms

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  • #1
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Homework Statement:
Suppose a given function f(x) and continuous x limit is given from say a to b.
Now if it is asked to say that sum over the values of f(x) over x between a and b.
Relevant Equations:
e.g
Can we write it as
$$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+.......f(b)=\int^b_a f(x)dx$$
e.g
Can we write it as
$$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+.......f(b)=\int^b_a f(x)dx$$......(?)
Although $$\int f(x)dx$$ given the area tracked by thr function with the x-axis between a and b
Thanks.
 
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Answers and Replies

  • #2
501
92
Homework Statement:: Suppose a given function f(x) and continuous x limit is given from say a to b.
Now if it is asked to say that sum over the values of f(x) over x between a and b.
Homework Equations:: e.g
Can we write it as
$$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+.......f(b)=\int^b_a f(x)dx$$

e.g
Can we write it as
$$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+.......f(b)=\int^b_a f(x)dx$$......(?)
Although $$\int f(x)dx$$ given the area tracked by thr function with the x-axis between a and b
Thanks.

I might be missing something here, but don't we need to multiply each of the [itex] f(x_i) [/itex] by a [itex] \delta x [/itex] to get an area? Otherwise, I thought that the area under a certain f(x) value for a continuous function is 0.

As to the rewritten version, what if [itex] b < 2a [/itex]. To me it would seem more intuitive to write out something like:
[tex] Integral = f(a)\delta x + f(a + \Delta x)\delta x + f(a + 2\Delta x)\delta x + f(a + 3\Delta x)\delta x + ... + f(b)\delta x [/tex]
 
  • #3
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Can we write it as
$$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+.......f(b)=\int^b_a f(x)dx$$
No.
As already mentioned by another poster, your coefficients are in the wrong places. The sum above is often written as
##f(a)+f(a + \Delta x)+f(a + 2\Delta )+f(a + 3\Delta ) + \dots + f(a + n \Delta x)##
where ##\Delta x = \frac {b - a} n##
This sum is called a Riemann sum, which is used to approximate a definite integral. For suitable functions (i.e., functions that are continuous on the interval [a, b]), the integral ##\int_a^b f(x) dx## is defined to be equal to the limit of the Riemann sum, as n goes to infinity.
 
  • #4
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5
$$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+.......f(b)=\int^b_a f(x)dx$$......(?)

The two sides will not be equal. As the user above said, the left side is a Riemann Sum and only used as approximation.
 
  • #5
34,983
6,738
As the user above said, the left side is a Riemann Sum
... that has no relationship with the integral on the right side.
 

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