7.1. Riemann Integral

Examples 7.1.13(a):

Find an upper and lower estimate for x sin(x) dx over the interval [0, 4].
Since | sin(x) | 1 we have:
-x x sin(x) x
for all x. By part 4 of the previous proposition we have:
- x dx x sin(x) dx x dx
for any a, b. But just as we computed x2 dx in a previous exercise we could show that:
x dx = 1/2 (b2 - a2)
(which would be a nice and simple exercise). Putting everything together we get that
-8 x sin(x) dx 8
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