Sigma is a discrete sum, a sum with steps. Eg. add the numbers from 1 to 10 or add the numbers 1/2, 1/4,...
A sigma always has a concept of a next thing to add, even if the list of things goes on forever.
An integral is a continuous summation. It is a summation in that we are adding up the area under the curve, for example, but it is continuous in that because we are adding things of arbitrary smallness it's not really possible to always point to the individual terms that are being added because they become some kind of continuous blur. Instead we use some mathematical technique (integration). But still it is a kind of summation.
Both sigma and pi bonds involve the sharing of a pair of electrons the difference is where the shared pair resides. In a sigma bond the electron pair mainly sits on the bond axis, the imaginary line between the atoms. In a pi bond the electrons sit above and below he bond axis and do not have any density along the axis.
Both kinds of integrals are essentially calculations of areas under curves. In a definite integral the surface whose area is to be calculated is planar. In a line integral the surface whose area to be calculated might occupy two or more dimensions. You might be interested in the animated diagrams in the wikipedia article for the line integral.
Constants can be integrated by multiplying the integrand(thing being integrated) times the difference between the starting point a and the ending point b i.e the integral from 0 to 10 would be 5,000,960 Depends. Are you referring to an indefinite integral (a plain ∫) or a definite integral? I'm not sure about this(i've only spent 2 days learning about integrals), but the indefinite integral of 500096 is 500096x+C. The definite integral of 500096 depends on what are your limits. Look at the answer above.
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