How do I find $\int_4^6 f(x) \,dx$ when the integrals $\int_0^6 f(x) \,dx$ and $\int_0^4 f(x) \,dx$ are given?

If $$\int_0^6 f(x) \,dx= 10$$ and $$\int_0^4 f(x) \,dx= 7$$

what is

$$\int_4^6 f(x) \,dx$$

?

1 Answer

Hint: $$\int_a^bf(x)dx+\int_b^cf(x)dx=\int_a^cf(x)dx$$ Plug in $a=0$, $b=4$, and $c=6$.

• @Donovan, one helpful thing to notice (at least it helps me) is that your first integral (0 to 6) overlaps both the second integral (0 to 4) AND the question integral (4 to 6). So, visualizing integrals as area under the curve f(x), you can ask the question: is Area(0 to 6) the same as Area(0 to 4) + Area(4 to 6) ? This is redundant with Akiva's answer, but easier for me to visualize this way. – jgreve Jan 13 '17 at 2:23