56. The region bounded by the graph of f(x) = 3x² + 1 and the x-axis on the interval [-1, 1].
56. The region bounded by the graph of f(x) = 3x² + 1 and the x-axis on the interval [-1, 1].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
(Question 56 and 89)
show all work thank you!
![**Approximating Areas with a Calculator**
Use a calculator and right Riemann sums to approximate the area of the given region. Present your calculations in a table showing the approximations for \( n = 10, 30, 60, \) and 80 subintervals. Make a conjecture about the limit of Riemann sums as \( n \to \infty \).
**Problems:**
55. The region bounded by the graph of \( f(x) = 12 - 3x^2 \) and the x-axis on the interval \([-1, 1]\).
56. The region bounded by the graph of \( f(x) = 3x^2 + 1 \) and the x-axis on the interval \([-1, 1]\).
57. The region bounded by the graph of \( f(x) = \frac{1 - \cos x}{2} \) and the x-axis on the interval \([-π, π]\).
58. The region bounded by the graph of \( f(x) = (2^x + 2^{-x}) \ln 2 \) and the x-axis on the interval \([-2, 2]\).
> **Instructions:**
>
> - Use the given functions and intervals to estimate the area under each curve.
> - Calculate Riemann sums using right endpoints for subintervals.
> - Analyze the results for different numbers of subintervals to observe how the approximations converge.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F627ffe10-a730-4e4c-b7e9-b306478e5985%2F2c8c3f20-b0b4-45de-a620-2f3d83655874%2F3840jt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Approximating Areas with a Calculator**
Use a calculator and right Riemann sums to approximate the area of the given region. Present your calculations in a table showing the approximations for \( n = 10, 30, 60, \) and 80 subintervals. Make a conjecture about the limit of Riemann sums as \( n \to \infty \).
**Problems:**
55. The region bounded by the graph of \( f(x) = 12 - 3x^2 \) and the x-axis on the interval \([-1, 1]\).
56. The region bounded by the graph of \( f(x) = 3x^2 + 1 \) and the x-axis on the interval \([-1, 1]\).
57. The region bounded by the graph of \( f(x) = \frac{1 - \cos x}{2} \) and the x-axis on the interval \([-π, π]\).
58. The region bounded by the graph of \( f(x) = (2^x + 2^{-x}) \ln 2 \) and the x-axis on the interval \([-2, 2]\).
> **Instructions:**
>
> - Use the given functions and intervals to estimate the area under each curve.
> - Calculate Riemann sums using right endpoints for subintervals.
> - Analyze the results for different numbers of subintervals to observe how the approximations converge.
Transcribed Image Text:**87–90. Graphing General Solutions**
Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition.
87. \( f'(x) = 2x - 5; \, f(0) = 4 \)
88. \( f'(x) = 3x^2 - 1; \, f(1) = 2 \)
89. \( f'(x) = 3x + \sin x; \, f(0) = 3 \)
90. \( f'(x) = \cos x - \sin x + 2; \, f(0) = 1 \)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
