Calculate Sc (6(x² - y)i + 5(y² + x)}) · dr
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Calculate the line integral
\[
\int_{C} \left( 6(x^2 - y)\vec{i} + 5(y^2 + x)\vec{j} \right) \cdot d\vec{r}
\]
given the following conditions:
**(a)** \( C \) is the circle \((x - 8)^2 + (y - 7)^2 = 16\) oriented counterclockwise.
\[
\int_{C} \left( 6(x^2 - y)\vec{i} + 5(y^2 + x)\vec{j} \right) \cdot d\vec{r} = \Box
\]
**(b)** \( C \) is the circle \((x - a)^2 + (y - b)^2 = R^2\) in the \( xy \)-plane oriented counterclockwise.
\[
\int_{C} \left( 6(x^2 - y)\vec{i} + 5(y^2 + x)\vec{j} \right) \cdot d\vec{r} = \Box
\]
**Explanation:**
- **Equation Description:**
- The problem requires calculating the line integral of a vector field given by \((6(x^2 - y)\vec{i} + 5(y^2 + x)\vec{j})\) over different circles in the plane.
- **Circle Details:**
- For part (a), the circle is centered at \((8, 7)\) with a radius of 4.
- For part (b), the circle is generically centered at \((a, b)\) with a radius \(R\).
- **Orientation:**
- Both circles are specified to be oriented counterclockwise, which impacts the direction of integration.
This exercise is designed to apply the concept of line integrals in vector calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff35f8f89-65fc-4b90-98e9-68f8eaa1b3da%2Ff90617a1-7334-4e92-a51c-5c3006a67dd2%2Fle3jhv2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Calculate the line integral
\[
\int_{C} \left( 6(x^2 - y)\vec{i} + 5(y^2 + x)\vec{j} \right) \cdot d\vec{r}
\]
given the following conditions:
**(a)** \( C \) is the circle \((x - 8)^2 + (y - 7)^2 = 16\) oriented counterclockwise.
\[
\int_{C} \left( 6(x^2 - y)\vec{i} + 5(y^2 + x)\vec{j} \right) \cdot d\vec{r} = \Box
\]
**(b)** \( C \) is the circle \((x - a)^2 + (y - b)^2 = R^2\) in the \( xy \)-plane oriented counterclockwise.
\[
\int_{C} \left( 6(x^2 - y)\vec{i} + 5(y^2 + x)\vec{j} \right) \cdot d\vec{r} = \Box
\]
**Explanation:**
- **Equation Description:**
- The problem requires calculating the line integral of a vector field given by \((6(x^2 - y)\vec{i} + 5(y^2 + x)\vec{j})\) over different circles in the plane.
- **Circle Details:**
- For part (a), the circle is centered at \((8, 7)\) with a radius of 4.
- For part (b), the circle is generically centered at \((a, b)\) with a radius \(R\).
- **Orientation:**
- Both circles are specified to be oriented counterclockwise, which impacts the direction of integration.
This exercise is designed to apply the concept of line integrals in vector calculus.
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