# If #int_1^3 \ f(x) \ dx = 5# and #int_3^8 \ f(x) \ dx = 10#, what is # int_1^8 \ f(x) \ dx #?

##### 2 Answers

# int_1^8 \ f(x) \ dx = 15 #

#### Explanation:

Using the properties of definite integral we have:

# int_a^c \ f(x) \ dx = int_a^b \ f(x) \ dx + int_b^c \ f(x) \ dx #

Hence we can write:

# int_1^8 \ f(x) \ dx = int_1^3 \ f(x) \ dx + int_3^8 \ f(x) \ dx #

# " " = 5 + 10 #

# " " = 15 #

#### Explanation:

We are asked to find

Just like how the area of any geometric shape can be found by breaking it into 2 pieces and adding together the two smaller areas, an integral

#int_a^cf(x)dx=int_a^bf(x)dx+int_b^cf(x)dx#

Here, we are given

#int_1^8f(x)dx=int_1^3f(x)dx+int_3^8f(x)dx#

#color(white)(int_1^8f(x)dx)=" "5" "+" "10#

#color(white)(int_1^8f(x)dx)=15# .