If f(x)=g(x)+7 for x in [3,5] then ∫ (5 on top, 3 on bottom) [ f(x)+g(x)]dx =
a 2∫ g(x)dx+7
b 2∫ g(x)dx+14
c 2∫ g(x)dx+28
d ∫ g(x)dx+7
e ∫ g(x)dx+14
all the ∫ has the five on top and 3 on bottom
please explain how to get that
thank you very much, i will vote for best answer when yahoo let me
1 Answer

Hi
Since f(x) = g(x) + 7 for x in [3, 5] and all the integrals we’re concerned with are only on the interval [3, 5], we have:
(3 to 5)∫ f(x) + g(x) dx
= (3 to 5)∫ g(x) + 7 + g(x) dx
= (3 to 5)∫ 2g(x) + 7 dx
= (3 to 5)∫ 2g(x) dx + (3 to 5)∫ 7 dx
(3 to 5)∫ 7 dx is the area under the constant function y = 7 from x = 3 to x = 5, which is just the area of a rectangle of width 5 – 3 = 2 and height 7, thus the area is 2*7 = 14, so we have:
(3 to 5)∫ 2g(x) dx + (3 to 5)∫ 7 dx
= (3 to 5)∫ 2g(x) dx + 14
= 2 * (3 to 5)∫ g(x) dx + 14 <— since we can pull out constants from the integral operator
So the answer is b.
I hope this helps!