Properties of definite integrals

Swapping the upper and lower limits flips the sign. If $\int_2^4 f(x)\,dx=-9$, then $\int_4^2 f(x)\,dx=9$.

When splitting at $b$, the upper limit of the first integral and the lower limit of the second must be the same $b$. Constants come outside, so if $\int_0^1 f(x)\,dx=2$, then $\int_0^1 3f(x)\,dx=3\cdot 2=6$.

Examples

1. 01What number goes in □?$\displaystyle \int_{1}^{4} f(x)\,dx = \int_{1}^{2} f(x)\,dx + \int_{\square}^{4} f(x)\,dx$AnswerHide

   Answer$\displaystyle 2$

   Steps

   1. $\displaystyle \int_{1}^{4} f(x)\,dx$$\displaystyle = \int_{1}^{2} f(x)\,dx + \int_{2}^{4} f(x)\,dx$
2. 02What is the value of this definite integral?$\displaystyle \int_{6}^{6} f(x)\,dx$AnswerHide

   Answer$\displaystyle 0$

   Steps

   1. $\displaystyle 6 \le x \le 6$$\displaystyle \Rightarrow \int_{6}^{6} f(x)\,dx = 0$
3. 03What is $\int_{4}^{2} f(x)\,dx$?$\displaystyle \int_{2}^{4} f(x)\,dx = -9$AnswerHide

   Answer$\displaystyle 9$
4. 04What is $\int_{0}^{1} 3f(x)dx$?$\displaystyle \int_{0}^{1} f(x)\,dx = 2$AnswerHide

   Answer$\displaystyle 6$
5. 05What number goes in □?$\displaystyle \int_{2}^{5} f(x)\,dx = \int_{2}^{3} f(x)\,dx + \int_{\square}^{5} f(x)\,dx$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle \int_{2}^{5} f(x)\,dx$$\displaystyle = \int_{2}^{3} f(x)\,dx + \int_{3}^{5} f(x)\,dx$
6. 06What number goes in □?$\displaystyle \int_{1}^{5} f(x)\,dx = \int_{1}^{3} f(x)\,dx + \int_{\square}^{5} f(x)\,dx$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle \int_{1}^{5} f(x)\,dx$$\displaystyle = \int_{1}^{3} f(x)\,dx + \int_{3}^{5} f(x)\,dx$

FAQ

Q1When splitting the interval, must $b$ lie between $a$ and $c$?AnswerHide

A

No. The formula holds whatever the order of $a$, $b$, $c$ (as long as $f(x)$ is continuous on an interval containing them). If $b$ is larger than $c$, the extra part cancels through $\int_b^c f(x)\,dx=-\int_c^b f(x)\,dx$.