Examine the graph the function $f(x) = 5 - x$ at right.

1. Find the area of the shaded region using geometry.

   Hint (a):

   Notice the familiar geometric shape.

   We will use the notation $A(f, 0 ≤ x ≤ 5)$ to represent "The area between the function and the $x$-axis" over the interval from $x = 0$ to $x=5$.

2. Notice that the line dips below the $x$-axis when $x > 5$. We will consider the area below the $x$ -axis as negative. Find $A(f, 0 ≤ x ≤ 7)$.

   Hint (b):

   Find the area of the negative region and combine
   it with the positive area found in part (a).

   More Help (b):

   $A (f, 0 ≤ x ≤ 5) + A (f, 5 ≤ x ≤7)$

3. Find $k$ if $A(f, 0 ≤ x ≤ k) = 0$. Show how you obtained your solution clearly and completely.

   Hint (c):

   $A (f, 0 ≤ x ≤ 5) + A (f, 5 ≤ x ≤ k)$

   Help:

   $[\text{area from part a}]+\frac{1}{2}(k-5)(-k+5)=\underline{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$