Question

if \\(\int_{1}^{7} f(x) \\, dx = 8.1\\) and \\(\int_{5}^{7} f(x) \\, dx = 5.6\\), find \\(\int_{1}^{5} f(x) \\, dx\\).

Explanation:

Step1: Recall integral interval property

$\int_{a}^{c} f(x)dx = \int_{a}^{b} f(x)dx + \int_{b}^{c} f(x)dx$
Rearrange to solve for $\int_{1}^{5} f(x)dx$:
$\int_{1}^{5} f(x)dx = \int_{1}^{7} f(x)dx - \int_{5}^{7} f(x)dx$

Step2: Substitute given values

Substitute $\int_{1}^{7} f(x)dx = 8.1$ and $\int_{5}^{7} f(x)dx = 5.6$:
$\int_{1}^{5} f(x)dx = 8.1 - 5.6$

Step3: Calculate the result

Perform the subtraction:
$8.1 - 5.6 = 2.5$

Answer:

2.5