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Answer :
To find the trapezoidal approximation of the integral [tex]\(\int_2^8 f(x) \, dx\)[/tex], we follow the trapezoidal rule using the given subintervals [tex]\([2,5]\)[/tex], [tex]\([5,7]\)[/tex], and [tex]\([7,8]\)[/tex]. The trapezoidal rule estimates the area under the curve by dividing it into trapezoids, then calculating the area of each trapezoid and summing these areas.
Here's a step-by-step explanation:
1. Determine the subintervals and their lengths:
- For [tex]\([2,5]\)[/tex], the length is [tex]\(5 - 2 = 3\)[/tex].
- For [tex]\([5,7]\)[/tex], the length is [tex]\(7 - 5 = 2\)[/tex].
- For [tex]\([7,8]\)[/tex], the length is [tex]\(8 - 7 = 1\)[/tex].
2. Calculate the area of each trapezoid:
The formula for the area of a trapezoid is [tex]\(\frac{1}{2} \times (\text{base}_2 - \text{base}_1) \times (\text{height}_1 + \text{height}_2)\)[/tex], where [tex]\(\text{base}_1\)[/tex] and [tex]\(\text{base}_2\)[/tex] are the x-values, and [tex]\(\text{height}_1\)[/tex] and [tex]\(\text{height}_2\)[/tex] are the corresponding values of [tex]\(f(x)\)[/tex].
- Area for [tex]\([2,5]\)[/tex]:
[tex]\[
\frac{1}{2} \times 3 \times (10 + 30) = \frac{1}{2} \times 3 \times 40 = 60
\][/tex]
- Area for [tex]\([5,7]\)[/tex]:
[tex]\[
\frac{1}{2} \times 2 \times (30 + 40) = \frac{1}{2} \times 2 \times 70 = 70
\][/tex]
- Area for [tex]\([7,8]\)[/tex]:
[tex]\[
\frac{1}{2} \times 1 \times (40 + 20) = \frac{1}{2} \times 1 \times 60 = 30
\][/tex]
3. Sum the areas of the trapezoids:
- The total trapezoidal approximation is:
[tex]\[
60 + 70 + 30 = 160
\][/tex]
Therefore, the trapezoidal approximation of [tex]\(\int_2^8 f(x) \, dx\)[/tex] is [tex]\(160\)[/tex].
The correct answer is [tex]\(\boxed{160}\)[/tex].
Here's a step-by-step explanation:
1. Determine the subintervals and their lengths:
- For [tex]\([2,5]\)[/tex], the length is [tex]\(5 - 2 = 3\)[/tex].
- For [tex]\([5,7]\)[/tex], the length is [tex]\(7 - 5 = 2\)[/tex].
- For [tex]\([7,8]\)[/tex], the length is [tex]\(8 - 7 = 1\)[/tex].
2. Calculate the area of each trapezoid:
The formula for the area of a trapezoid is [tex]\(\frac{1}{2} \times (\text{base}_2 - \text{base}_1) \times (\text{height}_1 + \text{height}_2)\)[/tex], where [tex]\(\text{base}_1\)[/tex] and [tex]\(\text{base}_2\)[/tex] are the x-values, and [tex]\(\text{height}_1\)[/tex] and [tex]\(\text{height}_2\)[/tex] are the corresponding values of [tex]\(f(x)\)[/tex].
- Area for [tex]\([2,5]\)[/tex]:
[tex]\[
\frac{1}{2} \times 3 \times (10 + 30) = \frac{1}{2} \times 3 \times 40 = 60
\][/tex]
- Area for [tex]\([5,7]\)[/tex]:
[tex]\[
\frac{1}{2} \times 2 \times (30 + 40) = \frac{1}{2} \times 2 \times 70 = 70
\][/tex]
- Area for [tex]\([7,8]\)[/tex]:
[tex]\[
\frac{1}{2} \times 1 \times (40 + 20) = \frac{1}{2} \times 1 \times 60 = 30
\][/tex]
3. Sum the areas of the trapezoids:
- The total trapezoidal approximation is:
[tex]\[
60 + 70 + 30 = 160
\][/tex]
Therefore, the trapezoidal approximation of [tex]\(\int_2^8 f(x) \, dx\)[/tex] is [tex]\(160\)[/tex].
The correct answer is [tex]\(\boxed{160}\)[/tex].
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