This question involves double integration of a function \( f(x, y) \) over a region \( R \). For each part of this question include a sketch of the region of integration \( R \).
(a)
\[
\begin{array}{l}
f(x, y)=x^{4} y+2 x y^{4} \\
R=\{(x, y): 0 \leq y \leq|x|,-1 \leq x \leq 1\}
\end{array}
\]
(i) Write down double definite integrals of \( f(x, y) \) over \( R \) in both ways (integrating with respect to \( x \) first, and integrating with respect to \( y \) first).
(ii) Evaluate either of the above integrals to verify that it is equal to \( \frac{1}{7} \).
(b) \( f(x, y)=(x+y) \)
\( R \) is the region bounded by the triangle with vertices \( (0,0),(1, \mathbb{1}) \), and \( (2,0) \).
Choose an order of integration (either \( d x d y \) or \( d y d x \) ), and evaluate the integral.
