SECTION 5.1 Quick Survey of Limits 251
Next, one knows that
∑n
i=1i^3 =^14 n^2 (n+ 1)^2 ; therefore,U(f;P) =4 n^2 (n+ 1)^2
n^4, L(f;P) =4 n^2 (n−1)^2
n^4.
Finally, we note that for any partition P′ of [0,2] 0 < L(f;P′) <
U(f;P′)<16, and so it is clear that GLB(U(f)) and LUB(L(f)) both
exist and that GLB(U(f))≥LUB(L(f)). Finally, for the partitionP
above we have
L(f;P)≤LUB(L(f))≤GLB(U(f))≤U(f;P).Therefore, we have
4 = limn→∞L(f;P)≤LUB(L(f))≤GLB(U(f))≤nlim→∞L(f;P) = 4,and so it follows that
∫ 2
0 x(^3) dx = 4.
For completeness’ sake, we present the following fundamental result
without proof.
Theorem. (Fundamental Theorem of Calculus) Assume that we are
given the function f defined on the interval [a,b]. If there exists a
differentiable functionF also defined on [a,b]and such that F′(x) =
f(x) for allx∈[a,b], thenf is Riemann integral on[a,b]and that
∫b
a f(x)dx = F(b)−F(a).
Exercises
- Letf andgbe functions such that
(a) f is defined in a punctured neighborhood ofa,