Answer
a) $0$
b) $0$
Work Step by Step
Recall: If $f(x)$ is above (or below) the $x-$axis for $a\leq x\leq b$ and $A$ is the area bounded by the curve of $f$ and the $x-axis$ for $a\leq x\leq b$, then $A=\int_a^bf(x) dx$ (or $A=-\int_a^b f(x)dx$).
Using this knowledge above, we have $\int_1^3f(x) dx=A$, $\int_3^4f(x)dx=-B$, and $\int_4^5 f(x)dx=C$.
Part a)
Using the properties for integrals,
$\int_1^4f(x)dx+\int_3^5 f(x)dx =(\int_1^3f(x)dx+\int_3^4f(x)dx)+(\int_3^4f(x)dx+\int_4^5f(x)dx)=(A+(-B))+((-B)+C)=(3+(-2))+(-2+1)=1+(-1)=0$
Part b)
Using the properties for integrals,
$\int_1^32f(x)dx+\int_3^56f(x)dx=2\int_1^3f(x)dx+6\int_3^5f(x)dx=2A+6(\int_3^4f(x)dx+\int_4^5f(x)dx)=2A+6((-B)+C)=2\cdot 3+6(-2+1)=6-6=0$
