the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) [√x³-8 dx, n = 10 (a) the Trapezoidal Rule (b) the Midpoint Rule (c) Simpson's Rule
the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) [√x³-8 dx, n = 10 (a) the Trapezoidal Rule (b) the Midpoint Rule (c) Simpson's Rule
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Topic: Numerical Integration: Trapezoidal Rule, Midpoint Rule, and Simpson's Rule**
In this exercise, we will approximate the integral of the function using three different numerical methods: the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule. The integral to approximate is:
\[ \int_{2}^{3} \sqrt{x^3 - 8} \, dx, \quad n = 10 \]
where \( n = 10 \) represents the number of subintervals.
### (a) The Trapezoidal Rule
\[ \boxed{} \]
### (b) The Midpoint Rule
\[ \boxed{} \]
### (c) Simpson's Rule
\[ \boxed{} \]
**Explanation:**
#### Trapezoidal Rule
The Trapezoidal Rule approximates the integral by dividing the interval into \( n \) subintervals and approximating the area under the curve as a series of trapezoids. The formula for the Trapezoidal Rule is:
\[
\int_a^b f(x) \, dx \approx \frac{b - a}{2n} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]
\]
#### Midpoint Rule
The Midpoint Rule approximates the integral by taking the value of the function at the midpoint of each subinterval. The formula for the Midpoint Rule is:
\[
\int_a^b f(x) \, dx \approx \frac{b - a}{n} \sum_{i=0}^{n-1} f \left( \frac{x_i + x_{i+1}}{2} \right)
\]
#### Simpson's Rule
Simpson's Rule provides a more accurate approximation by fitting parabolas through the function at intervals. The formula for Simpson's Rule is:
\[
\int_a^b f(x) \, dx \approx \frac{b - a}{3n} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right]
\]
All the above approximations require partitioning the interval \([2, 3]\) into \( 10 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5abad2ef-7d64-4544-b6f7-dd3dfbff74f8%2F348399b4-202b-4029-96da-e503f6a63ed9%2Fsml571_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Numerical Integration: Trapezoidal Rule, Midpoint Rule, and Simpson's Rule**
In this exercise, we will approximate the integral of the function using three different numerical methods: the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule. The integral to approximate is:
\[ \int_{2}^{3} \sqrt{x^3 - 8} \, dx, \quad n = 10 \]
where \( n = 10 \) represents the number of subintervals.
### (a) The Trapezoidal Rule
\[ \boxed{} \]
### (b) The Midpoint Rule
\[ \boxed{} \]
### (c) Simpson's Rule
\[ \boxed{} \]
**Explanation:**
#### Trapezoidal Rule
The Trapezoidal Rule approximates the integral by dividing the interval into \( n \) subintervals and approximating the area under the curve as a series of trapezoids. The formula for the Trapezoidal Rule is:
\[
\int_a^b f(x) \, dx \approx \frac{b - a}{2n} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]
\]
#### Midpoint Rule
The Midpoint Rule approximates the integral by taking the value of the function at the midpoint of each subinterval. The formula for the Midpoint Rule is:
\[
\int_a^b f(x) \, dx \approx \frac{b - a}{n} \sum_{i=0}^{n-1} f \left( \frac{x_i + x_{i+1}}{2} \right)
\]
#### Simpson's Rule
Simpson's Rule provides a more accurate approximation by fitting parabolas through the function at intervals. The formula for Simpson's Rule is:
\[
\int_a^b f(x) \, dx \approx \frac{b - a}{3n} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right]
\]
All the above approximations require partitioning the interval \([2, 3]\) into \( 10 \
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