Riemann Integration, Unbounded Function Usually, we understand that if a function $f\colon [a,b]\rightarrow \mathbb{R}$ is non-negative and integrable, then $\int_a^b f$ is the area under the graph of $f$. Also, the definition of Riemann integrable function considers only bounded functions (and also it is proved that a Riemann integrable on $[a,b]$ must be bounded. 
However, we can easily construct non-negative functions from $[a,b]$ to $\mathbb{R}$ such that the area under graph of $f$ is bounded. Can we conclude that the function is Riemann integrable? Where the (technical) difficulty arises to say that these unbounded functions, with bounded area, are Riemann integrable?
 A: Also if the function is bounded & this is continuous except infinitely many points hab=ving finitely many limit points. [If you know measure then the bounded function having points discontinuity with measure zero is also integrable.]
Now see integration is just the sum of the breaking pieces of the area under the curve. This breaking pieces are called the partition. Now if the function is unbounded then the however small the pieces are $\exists$ atleast one piece whose area is not bounded so the total sum goes to $\infty$.See the two images 

Adding some more details
What I have said previously see the definition of Riemann integrability. Here we have take the sum of the area of the partitions, but where f(1/n)=n the area is unbounded near the partition of zero. For one sheet of hyperbola if it is defined in $[a,b]$ then it is integrable[why??] & if it is defined on $\Bbb R^+$ then see the Riemann sum gets unbounded when $x \to \infty$. 
Definition of Riemann Integrability
Choose a real-valued function $ f$ which is defined on the interval $[a, b]$. The Riemann sum of $f$ with respect to the tagged partition $x_0,\ldots,x_n$ together with $t_0,\ldots,t_{n-1}$ is
$\sum_{i=0}^{n-1} f(t_i) (x_{i+1}-x_i)$.
Each term in the sum is the product of the value of the function at a given point, and the length of an interval. Consequently, each term represents the area of a rectangle with height $f(t_i)$ and width $x_{i+1}-x_i$. The Riemann sum is the signed area under all the rectangles.
