QuestionLet . Find the line integral of around the perimeter of the rectangle with corners , traversed in that order. line integral =
Studdy Solution
STEP 1
1. The vector field is defined and continuous over the region of interest.
2. The path of integration is the perimeter of the rectangle with corners , , , and .
3. The line integral is taken in the counterclockwise direction as specified by the order of the corners.
STEP 2
1. Parameterize each segment of the rectangle.
2. Compute the line integral over each segment.
3. Sum the contributions from all segments to find the total line integral.
STEP 3
Parameterize each segment of the rectangle:
- Segment 1: From to
$ \begin{align*}
x(t) &= 3, \\
y(t) &= t, \quad 0 \leq t \leq 5.
\end{align*}
\]
- Segment 2: From to
$ \begin{align*}
x(t) &= 3 - t, \\
y(t) &= 5, \quad 0 \leq t \leq 4.
\end{align*}
\]
- Segment 3: From to
$ \begin{align*}
x(t) &= -1, \\
y(t) &= 5 - t, \quad 0 \leq t \leq 5.
\end{align*}
\]
- Segment 4: From to
$ \begin{align*}
x(t) &= -1 + t, \\
y(t) &= 0, \quad 0 \leq t \leq 4.
\end{align*}
\]
STEP 4
Compute the line integral over each segment:
- Segment 1:
$ \int_{0}^{5} \left[ 5(3+t) \cdot 0 + 4 \sin(t) \cdot 1 \right] \, dt = \int_{0}^{5} 4 \sin(t) \, dt
\]
Evaluate:
$ \left[ -4 \cos(t) \right]_{0}^{5} = -4 \cos(5) + 4 \cos(0)
\]
- Segment 2:
$ \int_{0}^{4} \left[ 5(3+(5)) \cdot (-1) + 4 \sin(5) \cdot 0 \right] \, dt = \int_{0}^{4} -40 \, dt
\]
Evaluate:
$ -40t \bigg|_{0}^{4} = -160
\]
- Segment 3:
$ \int_{0}^{5} \left[ 5(-1+(5-t)) \cdot 0 + 4 \sin(5-t) \cdot (-1) \right] \, dt = \int_{0}^{5} -4 \sin(5-t) \, dt
\]
Let , then :
$ \int_{5}^{0} -4 \sin(u) \, (-du) = \int_{0}^{5} 4 \sin(u) \, du
\]
Evaluate:
$ \left[ -4 \cos(u) \right]_{0}^{5} = -4 \cos(5) + 4 \cos(0)
\]
- Segment 4:
$ \int_{0}^{4} \left[ 5(-1+t) \cdot 1 + 4 \sin(0) \cdot 0 \right] \, dt = \int_{0}^{4} (5t - 5) \, dt
\]
Evaluate:
$ \left[ \frac{5}{2}t^2 - 5t \right]_{0}^{4} = \left( \frac{5}{2}(16) - 20 \right) = 40 - 20 = 20
\]
STEP 5
Sum the contributions from all segments:
Simplify:
The line integral is:
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