Math Problem Statement
Let f be a continuous function on the interval [0,8] . If we use the Trapezoidal Rule with n=4 to approximate the integral ∫f(x)dx lower limit 0 upper limit 8, which of the following is the required approximation? Question 1Select one: (f(0)+f(2)+f(4)+f(6)+f(8))2 cross out (f(0)+2f(2)+2f(4)+2f(6)+f(8))1/2 cross out f(0)+2f(2)+2f(4)+2f(6)+f(8) cross out f(0)+2f(1)+2f(2)+2f(3)+2f(4)+2f(5)+2f(6)+2f(7)+f(8) cross out
Solution
To apply the Trapezoidal Rule for approximating the integral , we use the following formula:
where:
- is the number of subintervals (here ),
- is the width of each subinterval,
- are the points in the interval [0, 8].
For , the interval [0, 8] is divided into 4 subintervals, so each subinterval has width: Thus, the points where the function will be evaluated are:
Using the trapezoidal rule formula:
Simplifying:
Thus, the correct approximation is:
Answer: The third option.
Would you like further explanation or clarification on this? Here are some related questions for practice:
- What is the general formula for the Trapezoidal Rule?
- How does the Trapezoidal Rule compare with Simpson's Rule for approximating integrals?
- What happens to the accuracy of the approximation as increases?
- How do you determine the error in the Trapezoidal Rule approximation?
- Can the Trapezoidal Rule be used for improper integrals?
Tip: When approximating integrals using the Trapezoidal Rule, the more subintervals you use, the more accurate the approximation generally becomes.
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Math Problem Analysis
Mathematical Concepts
Numerical Integration
Trapezoidal Rule
Approximation Methods
Formulas
T_n = h/2 * (f(x_0) + 2 Σ f(x_i) + f(x_n))
Theorems
Trapezoidal Rule for numerical integration
Suitable Grade Level
Grades 11-12