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Answer :
To find the approximate area between the [tex]\(x\)[/tex]-axis and the function [tex]\(g(x)\)[/tex] from [tex]\(x = 10\)[/tex] to [tex]\(x = 16\)[/tex] using a trapezoidal sum with 3 equal subdivisions, follow the steps below:
1. Identify the Trapzoids' Width: First, calculate the width of each trapezoid. Here, we have the [tex]\(x\)[/tex]-values as 10, 12, 14, and 16. These values are spaced equally, so each interval's width [tex]\(h\)[/tex] is:
[tex]\[
h = \frac{16 - 10}{3} = 2
\][/tex]
2. Apply the Trapezoidal Rule: The trapezoidal rule formula for approximating the area under a curve is:
[tex]\[
\text{Area} = \frac{h}{2} \left( g(x_0) + 2g(x_1) + 2g(x_2) + \ldots + 2g(x_{n-1}) + g(x_n) \right)
\][/tex]
In our case, [tex]\(x_0 = 10\)[/tex], [tex]\(x_1 = 12\)[/tex], [tex]\(x_2 = 14\)[/tex], [tex]\(x_3 = 16\)[/tex], and [tex]\(g(x)\)[/tex] values are given as follows:
- [tex]\(g(10) = 5\)[/tex]
- [tex]\(g(12) = 1\)[/tex]
- [tex]\(g(14) = 7\)[/tex]
- [tex]\(g(16) = 7\)[/tex]
3. Plug in the Values: Substitute these values into the trapezoidal rule formula:
[tex]\[
\text{Area} = \frac{2}{2} \left( 5 + 2(1) + 2(7) + 7 \right)
\][/tex]
4. Calculate the Area:
- Calculate the expression inside the parentheses:
[tex]\[
5 + 2(1) + 2(7) + 7 = 5 + 2 + 14 + 7 = 28
\][/tex]
- Therefore, the total area is:
[tex]\[
\text{Area} = 1 \times 28 = 28
\][/tex]
Thus, the approximate area between the [tex]\(x\)[/tex]-axis and [tex]\(g(x)\)[/tex] from [tex]\(x = 10\)[/tex] to [tex]\(x = 16\)[/tex] using a trapezoidal sum with 3 equal subdivisions is 28.0 square units.
1. Identify the Trapzoids' Width: First, calculate the width of each trapezoid. Here, we have the [tex]\(x\)[/tex]-values as 10, 12, 14, and 16. These values are spaced equally, so each interval's width [tex]\(h\)[/tex] is:
[tex]\[
h = \frac{16 - 10}{3} = 2
\][/tex]
2. Apply the Trapezoidal Rule: The trapezoidal rule formula for approximating the area under a curve is:
[tex]\[
\text{Area} = \frac{h}{2} \left( g(x_0) + 2g(x_1) + 2g(x_2) + \ldots + 2g(x_{n-1}) + g(x_n) \right)
\][/tex]
In our case, [tex]\(x_0 = 10\)[/tex], [tex]\(x_1 = 12\)[/tex], [tex]\(x_2 = 14\)[/tex], [tex]\(x_3 = 16\)[/tex], and [tex]\(g(x)\)[/tex] values are given as follows:
- [tex]\(g(10) = 5\)[/tex]
- [tex]\(g(12) = 1\)[/tex]
- [tex]\(g(14) = 7\)[/tex]
- [tex]\(g(16) = 7\)[/tex]
3. Plug in the Values: Substitute these values into the trapezoidal rule formula:
[tex]\[
\text{Area} = \frac{2}{2} \left( 5 + 2(1) + 2(7) + 7 \right)
\][/tex]
4. Calculate the Area:
- Calculate the expression inside the parentheses:
[tex]\[
5 + 2(1) + 2(7) + 7 = 5 + 2 + 14 + 7 = 28
\][/tex]
- Therefore, the total area is:
[tex]\[
\text{Area} = 1 \times 28 = 28
\][/tex]
Thus, the approximate area between the [tex]\(x\)[/tex]-axis and [tex]\(g(x)\)[/tex] from [tex]\(x = 10\)[/tex] to [tex]\(x = 16\)[/tex] using a trapezoidal sum with 3 equal subdivisions is 28.0 square units.
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