Question

\[\int\limits_{1}^{5}f(x)dx = 12 , \int\limits_{4}^{5}f(x)dx = 3\]

find

\[\int\limits_{4}^{1} 2f(x)dx\]

I think the answer is

-(2)(9) = -18

but I dont know how to express it

Answer 1

\[\int\limits_{1}^{5}f(x)dx=\int\limits_{1}^{4}f(x)dx+\int\limits_{4}^{5}f(x)dx\]

Answer 2

-18 is correct

Answer 3

\[\int\limits_{1}^{5}f(x) - \int\limits_{5}^{4}f(x) = -\int\limits_{4}^{1}f(x) = -(12-3)\]

Answer 4

thus

\[\int\limits_{4}^{1}2(f(x))dx = 2(-9) = -18\]

Answer 5

Area under curve a->c =area a->b+area b->c
\[\int\limits_{a}^{c}f(x)dx=\int\limits_{a}^{b}f(x)dx+\int\limits_{b}^{c}f(x)dx\]