4.3 Definite Integral Rules and Area

Which of the following is NOT a definite integral property?

• $\int_a^{a\ }f\left(x\right)\ dx=0$

• $\int_a^b\ f\left(x\right)\ dx\ =\ \int_a^c\ f\left(x\right)\ dx\ +\int_c^b\ f\left(x\right)\ dx$

• $\int_b^a\ f\left(x\right)\ dx\ =\int_a^b\ f\left(x\right)\ dx$

• $\int_a^b\ kf\left(x\right)dx\ =\ k\int_a^b\ f\left(x\right)\ dx$

Which of the following definite integrals describes the area under the curve?

• $\int_0^4\ 4-2x\ dx$

• $\int_0^2\ 4-2x\ dx$

• $\int_2^0\ 4-2x\ dx$

• $\int_4^0\ 4-2x\ dx$

Sketch the region that describes the definite integral. Then, use geometry to evaluate the integral.
 $\int_0^4\ \frac{x}{2}\ dx=$  

• 2

• 4

• 8

• 16

Given the following values:
 $\int_2^4\ x^3\ dx=60$  ,  $\int_2^4\ x\ dx\ =6$  ,  $\int_2^4\ dx=2$  
Evaluate the integral  $\int_2^4\ 15\ dx=$  

• 0

• 15

• 30

• 90

Given the following values: 
 $\int_2^4\ x^3dx=60$  ,  $\int_2^4\ x\ dx=6$  ,  $\int_2^4\ dx=2$  
Evaluate the integral  $\int_2^4\left(x^3+4\right)dx=$  

• 68

• 64

• 224

• 220

Given  $\int_4^8\ f\left(x\right)\ dx=12$  and  $\int_4^8\ g\left(x\right)\ dx=5$ 
Evaluate  $\int_4^8\left[f\left(x\right)+g\left(x\right)\right]dx=$  
 

• -17

• 7

• -7

• 17

Given $\int_4^8\ f\left(x\right)\ dx=12$  and  $\int_4^8\ g\left(x\right)\ dx=5$  
Evaluate  $\int_4^8\left[f\left(x\right)-g\left(x\right)\right]dx=$  

• -7

• 7

• -17

• 17

Given $\int_4^8\ f\left(x\right)\ dx=12$  and  $\int_4^8\ g\left(x\right)dx=5$  
Evaluate  $\int_4^8\left[2f\left(x\right)-3g\left(x\right)\right]=$  

• -9

• 26

• -26

• 9

Given $\int_4^8\ f\left(x\right)\ dx=12\$  and  $\int_4^8\ g\left(x\right)\ dx=5$  
Evaluate  $\int_4^8\ 7f\left(x\right)dx=$  

• 84

• 56

• -84

• -56

Use geometry to find $\int_3^4\ 3f\left(x\right)dx=$  

• 2

• 4

• 6

• 8