Comparing 2 Kinds of Integrals - Help Understanding Algebraically

[balaaditya]

Hi all,

I need help.

What is the difference between

$$\int_{0}^b x^2 dx$$

with

$$lim_{b\rightarrow \ \infty} \int_{0}^b x^2 dx$$

?

Can someone please show me algebraically for its clarity?

I don’t understand of the 2nd integral equation means.
Especially the appears of limit as b approach to infinity.

Thanks in advance

 

[motornoob101]

one is a proper integral and the other one is an improper integral

 

[sutupidmath]

The first integral is a regular definite integral. It basically just measures the area that the parabola $$y=x^2$$ encloses with the x-axis from the point x=0 to another point x=b.


The second one is an improper integral. THati is:

$$\int_{0}^{\infty}x^2dx=\lim_{b\to\infty}\int_{0}^{b}x^2dx$$

SO it means that the function x^2 is unbounde from the above. And if that integral converges, than it means that we are calculating the area that the function $$f(x)=x^2$$ encloses with the x-axis from x=0 to infinity. In other words the area that the right wing of the parabola encloses with the x-axis.