To approximate the area between the curve [tex]\( f(x) = -2x^2 + 32x + 5 \)[/tex] and the [tex]\( x \)[/tex]-axis on the interval [tex]\([0, 16]\)[/tex] using 4 rectangles, we will use the left endpoint of each rectangle to determine the height.
### Step-by-Step Solution:
1. Determine the width of each rectangle:
The interval [tex]\([0, 16]\)[/tex] is divided into 4 equal parts (rectangles).
[tex]\[
\text{Width of each rectangle} = \frac{{b - a}}{n} = \frac{{16 - 0}}{4} = 4
\][/tex]
2. Identify the left endpoints of each rectangle:
We will use the left endpoints to determine the heights. The left endpoints for each of the 4 rectangles are:
- [tex]\( x_0 = 0 \)[/tex]
- [tex]\( x_1 = 4 \)[/tex]
- [tex]\( x_2 = 8 \)[/tex]
- [tex]\( x_3 = 12 \)[/tex]
3. Evaluate the function at each left endpoint:
We need to find the height of each rectangle by evaluating the function [tex]\( f(x) \)[/tex] at these points.
- [tex]\( f(0) = -2(0)^2 + 32(0) + 5 = 5 \)[/tex]
- [tex]\( f(4) = -2(4^2) + 32(4) + 5 = -2(16) + 128 + 5 = -32 + 128 + 5 = 101 \)[/tex]
- [tex]\( f(8) = -2(8^2) + 32(8) + 5 = -2(64) + 256 + 5 = -128 + 256 + 5 = 133 \)[/tex]
- [tex]\( f(12) = -2(12^2) + 32(12) + 5 = -2(144) + 384 + 5 = -288 + 384 + 5 = 101 \)[/tex]
4. Calculate the area of each rectangle:
The area of each rectangle is given by the width times the height.
- Area of the first rectangle: [tex]\(4 \times f(0) = 4 \times 5 = 20\)[/tex]
- Area of the second rectangle: [tex]\(4 \times f(4) = 4 \times 101 = 404\)[/tex]
- Area of the third rectangle: [tex]\(4 \times f(8) = 4 \times 133 = 532\)[/tex]
- Area of the fourth rectangle: [tex]\(4 \times f(12) = 4 \times 101 = 404\)[/tex]
5. Sum the areas of all rectangles to get the total approximate area:
[tex]\[
\text{Total approximate area} = 20 + 404 + 532 + 404 = 1360
\][/tex]
Thus, the approximate area between the curve [tex]\( f(x) = -2x^2 + 32x + 5 \)[/tex] and the [tex]\( x \)[/tex]-axis on the interval [tex]\([0, 16]\)[/tex] using 4 rectangles is:
[tex]\[
\boxed{1360} \, \text{square units}
\][/tex]
