If f(x)=g(x)+7 for x in [3,5] then ∫ (5 on top, 3 on bottom) [ f(x)+g(x)]dx =?

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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!

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