 #1
fishturtle1
 392
 81
 Homework Statement:
 Show ##\vert \int_{2\pi}^{2\pi} x^2 \sin^8(e^x) \vert \le \frac{16\pi^3}{3}##
 Relevant Equations:

From previous example, ##\int_{0}^{b} x^2 dx = \frac{b^3}{3}## where ##b > 0##.
##M(f, S) = \sup\lbrace f(x) : x \in S \rbrace##
##m(f, S) = \inf\lbrace f(x) : s \in S \rbrace##
##U(f) = \inf\lbrace M(f, P) : \text{P is a partition of [a, b]} \rbrace##
##L(f) = \sup\lbrace m(f, P) : \text{P is a partition of [a, b]} \rbrace##
We say ##f## is integrable on ##[a,b]## if ##U(f) = L(f)## in which case ##\int_a^b f = U(f)##.
Theorem 33.5: If ##f## is integrable on ##[a,b]##, then ##\vert f \vert## is integrable on ##[a,b]## and $$\vert \int_a^b f \vert \le \int_a^b \vert f \vert$$
Theorem 33.6. Let ##f## be a function defined on ##[a,b]##. If ##a < c < b## and ##f## is integrable on ##[a,c]## and ##[c,b]##, then ##f## is integrable on ##[a,b]## and
$$\int_a^b f = \int_a^c f + \int_c^b f$$
##\textbf{Attempt at solution:}## By theorem 33.5., $$\vert \int_{2\pi}^{2\pi} x^2\sin^8(e^x) dx \vert \le \int_{2\pi}^{2\pi} \vert x^2\sin^8(e^x) \vert dx $$ $$= \int_{2\pi}^{2\pi} \vert x^2 \vert \vert \sin^8(e^x) \vert dx$$ $$\le \int_{2\pi}^{2\pi} \vert x^2 \vert dx = \int_{2\pi}^{2\pi} x^2 dx$$
By theorem 33.6, we have
$$\int_{2\pi}^{2\pi} x^2 dx = \int_{2\pi}^0 x^2 dx + \int_0^{2\pi}x^2 dx$$
By example 2, we have $$\int_0^{2\pi} x^2 dx = \frac{8\pi^3}{3}$$
Now I try to mimic example 2 to evaluate the other integral, but i'm stuck: Let ##P = \lbrace 2\pi = t_0 < t_1 < \dots < t_n = 0 \rbrace## be a partition of ##[2\pi, 0]##. We have $$U(x^2, P) = \sum_{k=1}^{n} M(x^2, [t_{k1}, t_k])\cdot(t_k  t_{k1}) = \sum_{k=1}^n (t_{k1})^2\cdot(t_k  t_{k1})$$
Setting ##t_k = \frac{k(2\pi)}{n}## we get $$U(f, P) = \sum_{k=1}^n \frac{(k1)^24\pi^2}{n^2} \frac{(2\pi)}{n} = \frac{8\pi^3}{n^3}\sum_{k=1}^n (k1)^2$$
$$ = \frac{8\pi^3}{n^3} \cdot \frac 16 (n1)(n)(2(n1) + 1) = \frac{8\pi^3}{n^3}\cdot(2n^3  3n^2 + n) \le \frac{8\pi^3}{3}$$
I was expecting to get ##U(x^2, P) \ge \frac{8\pi^3}{3}## and then could somehow argue ##U(x^2) = \frac{8\pi^3}{3}## but i'm not sure where to go from here.
By theorem 33.6, we have
$$\int_{2\pi}^{2\pi} x^2 dx = \int_{2\pi}^0 x^2 dx + \int_0^{2\pi}x^2 dx$$
By example 2, we have $$\int_0^{2\pi} x^2 dx = \frac{8\pi^3}{3}$$
Now I try to mimic example 2 to evaluate the other integral, but i'm stuck: Let ##P = \lbrace 2\pi = t_0 < t_1 < \dots < t_n = 0 \rbrace## be a partition of ##[2\pi, 0]##. We have $$U(x^2, P) = \sum_{k=1}^{n} M(x^2, [t_{k1}, t_k])\cdot(t_k  t_{k1}) = \sum_{k=1}^n (t_{k1})^2\cdot(t_k  t_{k1})$$
Setting ##t_k = \frac{k(2\pi)}{n}## we get $$U(f, P) = \sum_{k=1}^n \frac{(k1)^24\pi^2}{n^2} \frac{(2\pi)}{n} = \frac{8\pi^3}{n^3}\sum_{k=1}^n (k1)^2$$
$$ = \frac{8\pi^3}{n^3} \cdot \frac 16 (n1)(n)(2(n1) + 1) = \frac{8\pi^3}{n^3}\cdot(2n^3  3n^2 + n) \le \frac{8\pi^3}{3}$$
I was expecting to get ##U(x^2, P) \ge \frac{8\pi^3}{3}## and then could somehow argue ##U(x^2) = \frac{8\pi^3}{3}## but i'm not sure where to go from here.