If $$\int_{-5}^{30} f(x) \, dx = 70$$ and $$\int_{15}^{30} f(x) \, dx = -30$$ then what is the result of the following? $$\int_{-5}^{15} f(x) \, dx$$
$\begingroup$
$\endgroup$
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
Please try to clean up your questions, but it would simply be $\int_{-5}^{15}f(x)dx$=$\int_{-5}^{30} f(x)dx $ - $\int_{15}^{30} f(x)dx $ =$(70-(-30))=100$
This is just using simple properties of the definite integral.
$\begingroup$
$\endgroup$
First note that $$ \int_{-5}^{30} f(x)dx=F(30)-F(-5)=70 $$ And $$ \int_{15}^{30} f(x)dx=F(30)-F(15)=-30 $$ So now we have $$ F(-5)=F(30)-70 $$ And $$ F(15)=F(30)+30 $$ Therefore $$ \int_{-5}^{15} f(x)dx=F(15)-F(-5) $$ $$ = F(30)+30- F(30)+70=30+70=100 $$