Answer
a. $-5$
b. $10$
c. $0$
d. $15$
Work Step by Step
a. By Th. 4.6, Additive lnterval Property,
$\displaystyle \int_{-1}^{1}f(x)dx = \displaystyle \int_{-1}^{0}f(x)dx+\int_{0}^{1}f(x)dx$
... subtract $\displaystyle \int_{0}^{1}f(x)dx$ from both sides ...
$\displaystyle \int_{-1}^{0}f(x)dx=\int_{-1}^{1}f(x)dx-\int_{0}^{1}f(x)dx$
$=0-5=-5$
b. $\displaystyle \quad \int_{b}^{a}f(x)dx=-\int_{a}^{b}f(x)dx$, so
$\displaystyle \int_{0}^{1}f(x)dx-\int_{1}^{0}f(x)dx=$
$\displaystyle =\int_{0}^{1}f(x)dx-(-\int_{0}^{1}f(x)dx)$
$=5-(-5)=10$
c.
$\displaystyle \int_{-1}^{1}3f(x)dx=$ (... Th. 4.7.1)
$3\displaystyle \int_{-1}^{1}f(x)dx=3(0) =0$
d.
$\displaystyle \int_{0}^{1}3f(x)dx =$(... Th. 4.7.1)
$=3\displaystyle \int_{0}^{1}f(x)dx=3 (5 ) =15$
