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 08f(x)dx\int_0^8 f(x)\,dx, we use the following formula:

Tn=h2(f(x0)+2i=1n1f(xi)+f(xn))T_n = \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)

where:

  • nn is the number of subintervals (here n=4n = 4),
  • hh is the width of each subinterval,
  • x0,x1,,xnx_0, x_1, \dots, x_n are the points in the interval [0, 8].

For n=4n = 4, the interval [0, 8] is divided into 4 subintervals, so each subinterval has width: h=804=2h = \frac{8 - 0}{4} = 2 Thus, the points where the function f(x)f(x) will be evaluated are: x0=0,x1=2,x2=4,x3=6,x4=8x_0 = 0, \quad x_1 = 2, \quad x_2 = 4, \quad x_3 = 6, \quad x_4 = 8

Using the trapezoidal rule formula:

T4=22(f(0)+2f(2)+2f(4)+2f(6)+f(8))T_4 = \frac{2}{2} \left( f(0) + 2f(2) + 2f(4) + 2f(6) + f(8) \right)

Simplifying:

T4=f(0)+2f(2)+2f(4)+2f(6)+f(8)T_4 = f(0) + 2f(2) + 2f(4) + 2f(6) + f(8)

Thus, the correct approximation is:

f(0)+2f(2)+2f(4)+2f(6)+f(8)f(0) + 2f(2) + 2f(4) + 2f(6) + f(8)

Answer: The third option.


Would you like further explanation or clarification on this? Here are some related questions for practice:

  1. What is the general formula for the Trapezoidal Rule?
  2. How does the Trapezoidal Rule compare with Simpson's Rule for approximating integrals?
  3. What happens to the accuracy of the approximation as nn increases?
  4. How do you determine the error in the Trapezoidal Rule approximation?
  5. 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