6-6 Applying Properties of Definite Integrals

antiderivative

indefinite integral

definite integral

derivative

calculating the area under a curve

 $\int_a^b\ f\left(x\right)\ dx=F\left(b\right)-F\left(a\right)$ 

integrating a function, with no +C

using rectangles to approximate the area under a curve

we didn't learn this!

 $\int_b^af\left(x\right)\ dx=-\int_a^bf\left(x\right)\ dx$  

 $\int_a^af\left(x\right)\ dx=0$  

 $\int_a^bf\left(x\right)\ dx+\int_b^cf\left(x\right)\ dx$  
 $=\int_a^cf\left(x\right)\ dx$  

 $\int_a^bf\left(x\right)\ dx-\int_b^cg\left(x\right)\ dx$  
 $=\int_a^c\left(f\left(x\right)-g\left(x\right)\right)\ dx$  

 $\int_a^bf\left(x\right)\ dx-\int_a^bg\left(x\right)\ dx$  
 $=\int_a^b\left(f\left(x\right)-g\left(x\right)\right)\ dx$  

 $f\left(x\right)$  is positive

 $f\left(x\right)$  is continuous

 $f\left(a\right)$  exists

 $f\left(b\right)$  exists

positive

negative

zero

positive

negative

zero

positive

negative

zero