Use 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.) ST 1 In(t) dt, n = 10 (a) the trapezoidal rule (b) the midpoint rule (c) Simpson's rule.
Use 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.) ST 1 In(t) dt, n = 10 (a) the trapezoidal rule (b) the midpoint rule (c) Simpson's rule.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter7: Integration
Section7.2: Substitution
Problem 1E: Integration by substitution is related to what differentiation method? What type of integrand...
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![**Approximating Definite Integrals Using Numerical Methods**
In this exercise, you will use the trapezoidal rule, the midpoint rule, and Simpson's rule to approximate the given integral with a specified value of \( n \). Round your answers to six decimal places.
### Problem Statement
Approximate the integral
\[
\int_{4}^{5} \frac{1}{\ln(t)} \, dt
\]
using \( n = 10 \).
#### Methods:
1. **The Trapezoidal Rule**:
\[
\text{Approximation} = \boxed{\ }
\]
2. **The Midpoint Rule**:
\[
\text{Approximation} = \boxed{\ }
\]
3. **Simpson's Rule**:
\[
\text{Approximation} = \boxed{\ }
\]
### Explanation of Methods
#### (a) The Trapezoidal Rule
The trapezoidal rule approximates the area under a curve by dividing it into trapezoids rather than rectangles. This method is beneficial for approximating the definite integral when the function is complex.
#### (b) The Midpoint Rule
The midpoint rule is another approach to approximating the area under a curve. It involves using the midpoint of each interval to estimate the height of the rectangle.
#### (c) Simpson's Rule
Simpson's rule provides an even more accurate approximation by fitting parabolas to the sections of the curve instead of using straight lines (as in the trapezoidal rule) or midpoints (as in the midpoint rule). This rule requires an even number of intervals.
To summarize, please go on to calculate the above integral using each method and round your results to six decimal places in the boxes provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2Fb9a34e07-27ae-4ac0-adc9-be1a056ea080%2F127jlde_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Approximating Definite Integrals Using Numerical Methods**
In this exercise, you will use the trapezoidal rule, the midpoint rule, and Simpson's rule to approximate the given integral with a specified value of \( n \). Round your answers to six decimal places.
### Problem Statement
Approximate the integral
\[
\int_{4}^{5} \frac{1}{\ln(t)} \, dt
\]
using \( n = 10 \).
#### Methods:
1. **The Trapezoidal Rule**:
\[
\text{Approximation} = \boxed{\ }
\]
2. **The Midpoint Rule**:
\[
\text{Approximation} = \boxed{\ }
\]
3. **Simpson's Rule**:
\[
\text{Approximation} = \boxed{\ }
\]
### Explanation of Methods
#### (a) The Trapezoidal Rule
The trapezoidal rule approximates the area under a curve by dividing it into trapezoids rather than rectangles. This method is beneficial for approximating the definite integral when the function is complex.
#### (b) The Midpoint Rule
The midpoint rule is another approach to approximating the area under a curve. It involves using the midpoint of each interval to estimate the height of the rectangle.
#### (c) Simpson's Rule
Simpson's rule provides an even more accurate approximation by fitting parabolas to the sections of the curve instead of using straight lines (as in the trapezoidal rule) or midpoints (as in the midpoint rule). This rule requires an even number of intervals.
To summarize, please go on to calculate the above integral using each method and round your results to six decimal places in the boxes provided.
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