Can you integrate on a scheme? As the question suggests, can you integrate on a scheme? How? I don't even know if this is even a well-posed question...
 A: Since this question is pretty open ended, I'll give a few answers for this question. Let's start with the down to earth measure theoretic setting: We need some set and equip this set with a sigma algebra, and attach to this sigma algebra a measure. For a general scheme, this question splits for the characteristic p and characteristic 0 setting.
For the characteristic 0 setting, you can look at the analytification of your scheme and try and topologize that. The classical construction of this is in Serre's GAGA. For a smooth projective variety $X$ over $\mathbb{C}$, you take the set $X(\mathbb{C})$ and give it the standard complex topology.
Another place this pops up is in the theory of smooth schemes. This setting is much more different because instead of looking at polynomials in $\mathbb{C}[x_1,\ldots,x_n]$ you are looking at the ring of smooth functions over $\mathbb{R}^n$. The set of Points $\text{Hom}_{\mathbb{R}-Alg}(C^\infty(\mathbb{R}^n),\mathbb{R})$ can be topologized with the standard euclidean topology. The set of points in this setting are just the evaluation morphisms, which is in bijection with $\mathbb{R}^n$. If you want to look at some smooth manifold in this setting, you look at the set of well defined evaluation morphisms over your ring. For example, if you look at $C^\infty(S^1)$ embedded into $C^\infty(\mathbb{R}^2)$ given by the implicit equation $x^2 + y^2 - 1$, then the only well defined points of $C^\infty(S^1)$ in your algebra of smooth functions on the plane are the evaluations on the points of your implicit equation. Again, you can go back to your standard measure theory and come up with some sort of integration theory.
Now, there is a generalization of measure theory on schemes over characteristic zero called motivic integration where you look at the space of arcs, that is $X(k[[t]])$, define some nice class of subsets of this space, and have your integral valued in the k-theory of varieties. This theory was inspired by p-adic integration which does a similar type of procedure.
Outside of the measure theory approach there is the homological/intersection theoretic approach of integration. Recall from algebraic topology that on smooth compact spaces you have poincare duality where
$$
H^k(X)\times H_{n-k}(X) \to \mathbb{Q}
$$
is a perfect pairing. If you fix a homology class $[Y]$ then, this gives you a linear map to $\mathbb{Q}$ giving you a notion of integration. This is more transparent in the differential geometric case since you are literally integrating differential forms over geometric chains, i.e., submanifolds. Since we are in the algebraic setting, you have to look at an analog of homology/cohomology. This is supposed to be given by looking at the chow ring and analogues of it for "not as nice" spaces. This is the analogue of calc III types of integrals.

For reference, you can look at 
Smooth Manifolds and Observables - Jet Nestruev
For more information about the functor of points/ algebraic approach to doing differential geometry. In addition, you can look at Loeser's
Seattle notes on Motivic Integration
for a nice overview of motivic integration and a discussion about p-adic integration.
