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Proof of: if f(x) > g(x) for a < x < b then a∫b f(x) dx > g(x).
Because we get f(x) ≥ g(x) then we knows that f(x) - g(x) ≥ 0 on a ≤ x ≤ b and therefore by Property 8 proved as above we know that,
a∫b f(x) - g(x) dx > 0
We know as well from Property 4,
a∫b f(x) - g(x) dx = a∫b f(x) dx - a∫b g(x) dx
Therefore, we then get,
a∫b f(x) dx - a∫b g(x) dx > 0
a∫b f(x) dx > a∫b g(x) dx
Proof of: If m ≤ f(x) ≤ M for a ≤ x ≤ b then m (b - a)≤ a∫b f(x) dx ≤ M (b - a).
Provide m ≤ f(x) ≤ M we can utilize Property 9 on each inequality to write,
a∫b m dx < a∫b f(x) dx ≤ a∫b M dx
So by Property 7 on the left and right integral to find,
m(b -a) < a∫b f(x) dx ≤ M (b -a)
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how do you regroup?
40.783-75
A water sprinkler operates in a circular pattern a distance of 10 ft. Evaluate the circumference of the spray? (π = 3.14) a. 31.4 ft b. 314 ft c. 62.8 ft d. 628 ft
20! 18!
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If the p th term of an AP is q and the q th term is p. P.T its n th term is (p+q-n). Ans: APQ a p = q a q = p a n = ? a + (p-1) d = q a + (q-1) d = p
/3x-5/-x-7=0
Doing these sums initially in this way helps children see why they carry over numbers to the next column. You may like to devise some related activities now. , EI) Give activ
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