Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Evaluate the line integral a) | 3x'yds, where Cis the curve r(t)=5ti + (3t-7)j. Ostsl. b) (xy + z)ds, where C is the helix r(t)=cos(t) i + sin(t)j + t k, 0sts3n C) F.dr. where the vector field F(x.y) = xfy'i -Wxj and C is the curve r(t)= 2t i + 7tj. Osts1. !!

Evaluate the line integral a) | 3x'yds, where Cis the curve r(t)=5ti + (3t-7)j. Ostsl. b) (xy + z)ds, where C is the helix r(t)=cos(t) i + sin(t)j + t k, 0sts3n C) F.dr. where the vector field F(x.y) = xfy'i -Wxj and C is the curve r(t)= 2t i + 7tj. Osts1. !!

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
Show all steps please
### Evaluating Line Integrals Here are a few problems related to the evaluation of line integrals in vector calculus: #### Problem a Evaluate the line integral \(\int_{C} 3x^3 y \, ds\), where \(C\) is the curve \( \mathbf{r}(t) = 5t \mathbf{i} + (3t - 7) \mathbf{j} \), \(0 \leq t \leq 1\). #### Problem b Evaluate the line integral \(\int_{C} (xy + z) \, ds\), where \(C\) is the helix \(\mathbf{r}(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k}\), \(0 \leq t \leq 3\pi \). #### Problem c Evaluate the line integral \(\int_{C} \mathbf{F} \cdot d\mathbf{r}\), where the vector field \(\mathbf{F}(x, y) = x^2 y^2 \mathbf{i} - y \sqrt{x} \mathbf{j}\) and \(C\) is the curve \(\mathbf{r}(t) = -2t^3 \mathbf{i} + 7t \mathbf{j}\), \(0 \leq t \leq 1\). ### Detailed Explanations: - **Vector Fields & Curves:** Each of these problems involves a specific type of curve \( \mathbf{r}(t) \) present within given limits for parameter \( t \). To evaluate these integrals, we typically need to parameterize the curve, find the differential element \( ds \) (or \( d\mathbf{r} \)), and substitute these into respective integrals. - **Helix Curve:** In Problem b, the helix is a three-dimensional curve defined by trigonometric functions which naturally wrap around a circular cylinder as \( t \) increases, making a helical shape. Would you like a step-by-step solution to any of these problems?
Transcribed Image Text:### Evaluating Line Integrals Here are a few problems related to the evaluation of line integrals in vector calculus: #### Problem a Evaluate the line integral \(\int_{C} 3x^3 y \, ds\), where \(C\) is the curve \( \mathbf{r}(t) = 5t \mathbf{i} + (3t - 7) \mathbf{j} \), \(0 \leq t \leq 1\). #### Problem b Evaluate the line integral \(\int_{C} (xy + z) \, ds\), where \(C\) is the helix \(\mathbf{r}(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k}\), \(0 \leq t \leq 3\pi \). #### Problem c Evaluate the line integral \(\int_{C} \mathbf{F} \cdot d\mathbf{r}\), where the vector field \(\mathbf{F}(x, y) = x^2 y^2 \mathbf{i} - y \sqrt{x} \mathbf{j}\) and \(C\) is the curve \(\mathbf{r}(t) = -2t^3 \mathbf{i} + 7t \mathbf{j}\), \(0 \leq t \leq 1\). ### Detailed Explanations: - **Vector Fields & Curves:** Each of these problems involves a specific type of curve \( \mathbf{r}(t) \) present within given limits for parameter \( t \). To evaluate these integrals, we typically need to parameterize the curve, find the differential element \( ds \) (or \( d\mathbf{r} \)), and substitute these into respective integrals. - **Helix Curve:** In Problem b, the helix is a three-dimensional curve defined by trigonometric functions which naturally wrap around a circular cylinder as \( t \) increases, making a helical shape. Would you like a step-by-step solution to any of these problems?
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Point Estimation, Limit Theorems, Approximations, and Bounds
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS