Certainly! Let's go through the problem step by step. --- ### **Step 1: Understanding the Table and Formula** The table gives values for \( f(x) \) at certain points. The question asks for the value of the definite integral: \[ \int_{}^{4} f(x) \, dx \] using the **trapezoidal rule** with the values from the table. The trapezoidal rule formula for \( n \) subintervals (here, 4 intervals: [,1], [1,2], [2,3], [3,4]) is: \[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2}\left[f(x_) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)\right] \] where \( h = 1 \) (since the x-values go from to 4 in steps of 1). **Values from table:** - \( f() = 1 \) - \( f(1) = 4 \) - \( f(2) = 2 \) - \( f(3) = 3 \) - \( f(4) = 1 \) --- ### **Step 2: Applying the Trapezoidal Rule** Plug in the values: \[ \int_{}^{4} f(x) \, dx \approx \frac{1}{2}\left[1 + 2(4) + 2(2) + 2(3) + 1\right] \] Calculate inside the brackets: - \( 1 + 2(4) = 1 + 8 = 9 \) - \( 2(2) = 4 \) - \( 2(3) = 6 \) - Add all: \( 9 + 4 + 6 + 1 = 20 \) Now multiply by \( \frac{1}{2} \): - \( \frac{1}{2} \times 20 = 10 \) --- ### **Final Answer** \[ \boxed{10} \] This is the approximate value of the integral using the trapezoidal rule.

Certainly! Let's solve the problem step by step, addressing all parts clearly. --- ## **Step 1: Understanding the Data** We have a table of values for \( f(x) \) at specific points, and we need to estimate the definite integral: \[ \int_0^4 f(x) \, dx \] using the **trapezoidal rule**. ### **Trapezoidal Rule Formula** The trapezoidal rule estimates the area under a curve by dividing it into \( n \) subintervals. The formula is: \[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2}\left[f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n)\right] \] where: - \( h \) is the width of each subinterval - \( n \) is the number of subintervals ### **Values from the Table** Given \( f(0) = 1 \), \( f(1) = 4 \), \( f(2) = 2 \), \( f(3) = 3 \), \( f(4) = 1 \): - \( f(x_0) = f(0) = 1 \) - \( f(x_1) = f(1) = 4 \) - \( f(x_2) = f(2) = 2 \) - \( f(x_3) = f(3) = 3 \) - \( f(x_4) = f(4) = 1 \) The width \( h \) is \( 1 \) since we are going from \( 0 \) to \( 4 \) in increments of \( 1 \). --- ## **Step 2: Applying the Trapezoidal Rule** Now we can substitute the values into the trapezoidal rule formula: \[ \int_0^4 f(x) \, dx \approx \frac{1}{2}\left[ f(0) + 2f(1) + 2f(2) + 2f(3) + f(4) \right] \] Substituting the values: \[ \int_0^4 f(x) \, dx \approx \frac{1}{2}\left[ 1 + 2(4) + 2(2) + 2(3) + 1 \right] \] Calculating inside the brackets: 1. Calculate \( 2(4) = 8 \) 2. Calculate \( 2(2) = 4 \) 3. Calculate \( 2(3) = 6 \) Now, sum them up: \[ 1 + 8 + 4 + 6 + 1 = 20 \] Now multiply by \( \frac{1}{2} \): \[ \frac{1}{2} \times 20 = 10 \] --- ## **Final Answer** The approximate value of the integral using the trapezoidal rule is: \[ \boxed{10} \]

Sure! Let's solve parts (a), (b), and (c) step by step. --- # **Part (a): Understanding the Problem** Given: - Values from a table for the function \( f(x) \) at specific points. - We need to estimate the definite integral: \[ \int_0^4 f(x) \, dx \] using the **trapezoidal rule**. ### **Trapezoidal Rule Formula** The trapezoidal rule is given by: \[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2}\left[f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n)\right] \] where: - \( h \) is the width of each subinterval. - \( n \) is the total number of intervals. ### **Values from the Table** Values for \( f(x) \): - \( f(0) = 1 \) - \( f(1) = 4 \) - \( f(2) = 2 \) - \( f(3) = 3 \) - \( f(4) = 1 \) Here, \( h = 1 \) (as the intervals are from \( 0 \) to \( 4 \) in steps of \( 1 \)). --- # **Part (b): Applying the Trapezoidal Rule** Substituting the values into the trapezoidal rule formula: \[ \int_0^4 f(x) \, dx \approx \frac{1}{2}\left[ f(0) + 2f(1) + 2f(2) + 2f(3) + f(4) \right] \] ### **Calculating Each Term** 1. \( f(0) = 1 \) 2. \( 2f(1) = 2 \times 4 = 8 \) 3. \( 2f(2) = 2 \times 2 = 4 \) 4. \( 2f(3) = 2 \times 3 = 6 \) 5. \( f(4) = 1 \) ### **Summing the Values** Now, sum these values: \[ 1 + 8 + 4 + 6 + 1 = 20 \] ### **Final Calculation** Now multiply by \( \frac{1}{2} \): \[ \frac{1}{2} \times 20 = 10 \] --- # **Part (c): Final Answer** The approximate value of the integral using the trapezoidal rule is: \[ \boxed{10} \] This concludes the solution to parts (a), (b), and (c).