QuestionIn Exercises 57-60, calculate the integral, assuming that $$\int_{0}^{1} f(x) d x=1, \quad \int_{0}^{2} f(x) d x=4, \quad \int_{1}^{4} f(x) d x=7$$
57. $\int_{0}^{4} f(x) d x$
58. $\int_{1}^{2} f(x) d x$

Studdy Solution
Substitute $x = 3$ back into the equation for $\int_{2}^{4} f(x) \, dx$:
$\int_{2}^{4} f(x) \, dx = 7 - 3 = 4$
Now, substitute back to find $\int_{0}^{4} f(x) \, dx$:
$\int_{0}^{4} f(x) \, dx = 4 + 4 = 8$
The values of the integrals are: - $\int_{0}^{4} f(x) \, dx = \boxed{8}$ - $\int_{1}^{2} f(x) \, dx = \boxed{3}$