1) $\int_Df(x)dx = \int_a^bf(x)dx$, where $x\longmapsto f(x)$. Gives the area under the curve between $a$ and $b$.
2) $\iint_Sf(x,y)dS = \int_{y_1}^{y_2}\int_{x_1}^{x_2}f(x,y)dxdy$, where $(x,y)\longmapsto f(x,y)$. Gives the area under the surface in the region $S$.
3) $\iiint_Vf(x,y)dV = \int_{z_1}^{z_2}\int_{y_1}^{y_2}\int_{x_1}^{x_2}f(x,y)dxdydz$, where $(x,y,z)\longmapsto f(x,y,z)$. Gives the area under the surface, as above.
4) $\int_D\underline{F}\cdot d\underline{r} = \int_a^b\underline{F}(\underline{r}(t))\frac{d\underline{r}}{dt}dt$. Where $\underline{F}$ is a vector field and $\underline{r} is a position vector. Gives the amount work done along a path with a certain vector field.
5) Integrals in complex analysis along a contour.
Now why the first three are different from 4 and 5? In the first three they give the area under the curve, in the 4th gives some kind of "density times length". The weird thing is that the integral along a closed path is equal to zero (in 4) only if the vector field is conservative, however it is always equal to zero in (5) no matter the vector field.
Why are all these integrals so different?
Correction:
and so on. Note that every integral here can be can be interpreted as density times generalized length.
The point is that all of these integrals can have multiple meanings. I certainly don't even claim that my two choices in each of the above are the only interpretations of those integrals. Integration is a very robust tool that can serve many, many different purposes.