Question

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

Answer 1

`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`

Answer 2

`int_1^8f(x)dx=15`.

Explanation:

We are asked to find `int_1^8f(x)dx,` which can be thought of as the total area between `f(x)` and the `x`-axis, from `x=1` on the left to `x=8` on the right.

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` over an interval `[a,c]` can be found by breaking it into two smaller intervals, `[a,b]` and `[b,c]`, and adding together their values. In other words, for any `b in (a,c):`

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

Here, we are given `int_1^3f(x)dx=5` and `int_3^8f(x)dx=10.` So, we can compute:

`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`.