Properties of Integrals � The symbol originated as a stylized letter S; in French, they call integrals sums. We know from our discussion of Riemann sums that deﬁnite integrals are just limits of sums. Because of this, it’s not surprising that: 1. The integral of a sum is the sum of the integrals: � b � b (f (x) + g(x)) dx = � f (x) dx + a b g(x) dx. a a 2. We can factor out a constant multiple: b � b � cf (x) dx = c f (x) dx a (c constant) a (don’t try to factor out a non-constant function!) 3. We can combine deﬁnite integrals. If a < b < c then: � b � f (x) dx + a c � b a c f (x) dx = f (x) dx a b c Figure 1: Combining two areas under a curve 4. �a a f (x) dx = 0 5. This statement gives us some freedom in choosing limits of integration and allows us to remove the condition that a < b < c from property (3): � b � f (x) dx = − a a f (x) dx. b This makes sense; F (b) − F (a) = −(F (a) − F (B)). 6. (Estimation) If f (x) ≤ g(x) and a < b, then: � b � f (x) dx ≤ a b g(x) dx. a In other words, if I’m going more slowly than you then you go further than I do. Caution: this only works if a < b. 7. (Change of Variables or “Substitution”) In� indeﬁnite integrals, if u = � u(x) then du = u� (x) dx and g(u)du = g(u(x))u� (x) dx. To adapt this to deﬁnite integrals we need to know what happens to our limits of integration; it turns out that the answer is very simple. � u2 � x2 g(u)du = g(u(x))u� (x) dx, u1 x1 where u1 = u(x1 ) and u2 = u(x2 ). This is true if u is always increasing or always decreasing on x1 < x < x2 ; in other words, if u� does not change sign. (If u� does change sign you must break the integral into pieces; we’ll see an example of this later.)