1. If C₁ is the segment from (x₁, y₁) to (x2, y2), find Ja x dy - y dx in terms of the four coordinates of these two points. 2. If the vertices of a polygon (with nonintersecting sides) are (x₁, y₁), (x2, y2),..., (xn, Yn) in counterclockwise order, show that the area of the polygon is Area = [(*192 - x2Y1) + (X2Y3 — X3Y2) + … + (xn−1Yn — XnYn−1) + (XnY1 − X1 Yn )]. 3. Use the shoelace formula to find the point D on the line y = 8 so that A(1, 4), B(4, 1), C(11, 6), and D form a quadrilateral having area 30. Remark: The area formula above is called the Shoelace Formula. You can search online for a way to visualize the formula; this visualization explains the name. The formula can also be written as follows. x1 X2 X2 X3 x1 Area = + In Yn 21||0 ≤ 0 ≤ 0≤ 2π Y1 Y2 Y2 Y3 Yı +...+
1. If C₁ is the segment from (x₁, y₁) to (x2, y2), find Ja x dy - y dx in terms of the four coordinates of these two points. 2. If the vertices of a polygon (with nonintersecting sides) are (x₁, y₁), (x2, y2),..., (xn, Yn) in counterclockwise order, show that the area of the polygon is Area = [(*192 - x2Y1) + (X2Y3 — X3Y2) + … + (xn−1Yn — XnYn−1) + (XnY1 − X1 Yn )]. 3. Use the shoelace formula to find the point D on the line y = 8 so that A(1, 4), B(4, 1), C(11, 6), and D form a quadrilateral having area 30. Remark: The area formula above is called the Shoelace Formula. You can search online for a way to visualize the formula; this visualization explains the name. The formula can also be written as follows. x1 X2 X2 X3 x1 Area = + In Yn 21||0 ≤ 0 ≤ 0≤ 2π Y1 Y2 Y2 Y3 Yı +...+
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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Please provide a clear and complete solution
![1. If C₁ is the segment from (x₁, y₁) to (x2, y2), find
Ja
x dy - y dx in terms of the four coordinates of these two points.
2. If the vertices of a polygon (with nonintersecting sides) are (x₁, y₁), (x2, y2),..., (xn, Yn) in counterclockwise order, show
that the area of the polygon is
Area = [(*192 - x2Y1) + (X2Y3 — X3Y2) + … + (xn−1Yn — XnYn−1) + (XnY1 − X1 Yn )].
3. Use the shoelace formula to find the point D on the line y = 8 so that A(1, 4), B(4, 1), C(11, 6), and D form a quadrilateral
having area 30.
Remark: The area formula above is called the Shoelace Formula. You can search online for a way to visualize the formula; this
visualization explains the name. The formula can also be written as follows.
x1 X2
X2 X3
x1
Area =
+
In
Yn
21||0 ≤
0 ≤ 0≤ 2π
Y1 Y2
Y2 Y3
Yı
+...+](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60490751-1c6c-473c-844b-470bbbb67b75%2F5125d143-6a30-4818-bc7a-b6ab7390441f%2Flexuc2q_processed.png&w=3840&q=75)
Transcribed Image Text:1. If C₁ is the segment from (x₁, y₁) to (x2, y2), find
Ja
x dy - y dx in terms of the four coordinates of these two points.
2. If the vertices of a polygon (with nonintersecting sides) are (x₁, y₁), (x2, y2),..., (xn, Yn) in counterclockwise order, show
that the area of the polygon is
Area = [(*192 - x2Y1) + (X2Y3 — X3Y2) + … + (xn−1Yn — XnYn−1) + (XnY1 − X1 Yn )].
3. Use the shoelace formula to find the point D on the line y = 8 so that A(1, 4), B(4, 1), C(11, 6), and D form a quadrilateral
having area 30.
Remark: The area formula above is called the Shoelace Formula. You can search online for a way to visualize the formula; this
visualization explains the name. The formula can also be written as follows.
x1 X2
X2 X3
x1
Area =
+
In
Yn
21||0 ≤
0 ≤ 0≤ 2π
Y1 Y2
Y2 Y3
Yı
+...+
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