Expressing an Integral as a sum of terms

[Apashanka]

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.

 

[Master1022]

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 $f(x_i)$ by a $\delta x$ 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 $b < 2a$. To me it would seem more intuitive to write out something like:
$$Integral = f(a)\delta x + f(a + \Delta x)\delta x + f(a + 2\Delta x)\delta x + f(a + 3\Delta x)\delta x + ... + f(b)\delta x$$

 

[Mark44]

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.

 

[currently]

$$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+...f(b)=\int^b_a f(x)dx$$...(?)

The two sides will not be equal. As the user above said, the left side is a Riemann Sum and only used as approximation.

 

[Mark44]

As the user above said, the left side is a Riemann Sum

... that has no relationship with the integral on the right side.