### Home > PC3 > Chapter 10 > Lesson 10.1.2 > Problem10-28

10-28.

Consider the functions $f(x) = x^{2} + 2$ and $g(x) = x^3$

The graph of an even function is symmetric across the $y$-axis.
The graph of an odd function has rotational symmetry about the origin.

1. Is $f$ even, odd, or neither? Justify your answer algebraically.

2. Is $g$ even, odd, or neither? Justify your answer algebraically.

3. Approximate the area under $y = f(x)$ over the interval $−5 ≤ x ≤ 5$. Clearly show your method.

Choose a number of rectangles, say $20$. Then each rectangle will have a width of $0.5$. The height will be $f(−5 + 0.5n)$. Calculate:

$\displaystyle \sum _ {n=0}^{19}0.5f(-5+0.5n)\:\text{ OR }\:\displaystyle \sum _ {n=1}^{20}0.5f(-5+0.5n)$

4. Approximate the area under $y = g(x)$ over the interval $−5 ≤ x ≤ 5$. Clearly show your method.

Sketch the curve and then sketch some rectangles in the given interval. What do you notice?