7.4 A arc length
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An arc can be approximated by dividing it into many small line segments and calculating the distance between each point using the distance formula. A rectifiable curve is one that has a finite arc length and is continuous with a smooth graph. The arc length of a curve y=f(x) between values a and b is calculated by taking the integral from a to b of the square root of 1 plus the derivative of f with respect to x, squared. Similarly, for a curve given by x=g(y), the arc length is calculated by taking the integral from c to d of the square root of 1 plus the derivative of g with respect to y, squared.
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(ii) Major & Minor axis : The straight line AA is called major axis and BB is called minor axis. The major and minor axis taken together are called the principal axes and its length will be given by
Length of major axis 2a Length of minor axis 2b
(iii) Centre : The point which bisect each chord of an ellipse is called centre (0,0) denoted by 'C'.
(iv) Directrix : ZM and Z M are two directrix and their equation are x= a/e and x = – a/e.
(v) Focus : S (ae, 0) and S (–ae,0) are two foci of an ellipse.
(vi) Latus Rectum : Such chord which passes through either focus and perpendicular to the major axis is called its latus rectum.
Length of Latus Rectum :
If L is (ae, 𝑙 ) then 2𝑙 is the length of
SP+S'P=
{(h ae)2 k 2} +
= 2a
Latus Rectum.
Length of Latus rectum is given by
2b2
.
h2(1– e2) + k2 = a2(1– e2)
Hence Locus of P(h, k) is given by. x2(1– e2) + y2 = a2(1– e2)
2
a
(vii) Relation between constant a, b, and e
a 2 b2
b2 = a2(1– e2) e2 =
a 2
x2
a 2 +
y
a 2 (1 e 2 ) = 1
e =
a 2
Result :
Major Axis
(a) Centre C is the point of intersection of the axes of an ellipse. Also C is the mid point of AA.
(b) Another form of standard equation of ellipse
x 2 y2
a 2 b2
1 when a < b.
Directrix Minor Axis Directrix x = -a/e x = a/e
Let us assume that a2(1– e2 )= b2
The standard equation will be given by
x2 y2
a2 b2
2.1.1 Various parameter related with standard ellipse :
In this case major axis is BB= 2b which is along y- axis and minor axis is AA= 2a along x- axis. Focus S(0,be) and S(0,–be) and directrix are y = b/e and y = –b/e.
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Assignments
These assignments are keyed to Edition 7E of James Stewart’s “Calculus” (Early Transcendentals)
Assignment 1. Spheres and Other Surfaces
Read 12.1 - 12.2 and 12.6
You should be able to do the following problems:
Section 12.1/Problems 11 - 18, 20 - 22 Section 12.6/Problems 1 - 48
Hand in the following problems:
1. The following equation describes a sphere. Find the radius and the coordinates of the center.
x2 + y2 + z2 = 2(x + y + z) + 1
2. A particular sphere with center (−3, 2, 2) is tangent to both the xy-plane and the xz-plane.
It intersects the xy-plane at the point (−3, 2, 0). Find the equation of this sphere.
3. Suppose (0, 0, 0) and (0, 0, −4) are the endpoints of the diameter of a sphere. Find the
equation of this sphere.
4. Find the equation of the sphere centered around (0, 0, 4) if the sphere passes through the
origin.
5. Describe the graph of the given equation in geometric terms, using plain, clear language:
z =
√
1 − x2 − y2
Sketch each of the following surfaces
6. z = 2 − 2
√
x2 + y2
7. z = 1 − y2
8. z = 4 − x − y
9. z = 4 − x2 − y2
10. x2 + z2 = 16
Assignment 2. Dot and Cross Products
Read 12.3 and 12.4
You should be able to do the following problems:
Section 12.3/Problems 1 - 28 Section 12.4/Problems 1 - 32
Hand in the following problems:
1. Let u⃗ =
⟨
0, 1
2
,
√
3
2
⟩
and v⃗ =
⟨√
2,
√
3
2
, 1
2
⟩
a) Find the dot product b) Find the cross product
2. Let u⃗ = j⃗ + k⃗ and v⃗ = i⃗ +
√
2 j⃗.
a) Calculate the length of the projection of v⃗ in the u⃗ direction.
b) Calculate the cosine of the angle between u⃗ and v⃗
3. Consider the parallelogram with the following vertices:
(0, 0, 0) (0, 1, 1) (1, 0, 2) (1, 1, 3)
a) Find a vector perpendicular to this parallelogram.
b) Use vector methods to find the area of this parallelogram.
4. Use the dot product to find the cosine of the angle between the diagonal of a cube and one of
its edges.
5. Let L be the line that passes through the points (0, −
√
3 , −1) and (0,
√
3 , 1). Let θ be the
angle between L and the vector u⃗ = 1√
2
⟨0, 1, 1⟩. Calculate θ (to the nearest degree).
Assignment 3. Lines and Planes
Read 12.5
You should be able to do the following problems:
Section 12.5/Problems 1 - 58
Hand in the following problems:
1a. Find the equation of the line that passes through (0, 0, 1) and (1, 0, 2).
b. Find the equation of the plane that passes through (1, 0, 0) and is perpendicular to the line in
part (a).
2. The following equation describes a straight line:
r⃗(t) = ⟨1, 1, 0⟩ + t⟨0, 2, 1⟩
a. Find the angle between the given line and the vector u⃗ = ⟨1, −1, 2⟩.
b. Find the equation of the plane that passes through the point (0, 0, 4) and is perpendicular to
the given line.
3. The following two lines intersect at the point (1, 4, 4)
r⃗ = ⟨1, 4, 4⟩ + t⟨0, 1, 0⟩ r⃗ = ⟨1, 4, 4⟩ + t⟨3, 5, 4⟩
a. Find the angle between the two lines.
b. Find the equation of the plane that contains every point o ...
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The document discusses methods for finding the volume of solids of revolution using integration.
The disk method involves slicing a solid of revolution into thin cross-sectional disks and adding up their volumes. The radius of each disk is given by the function defining the region's boundary, and the volume of each disk is πr^2h, where h is the width of the disk.
The washer method is similar but used when the region has a hole cut out. Each slice has the shape of a washer rather than a solid disk.
Examples are given revolving regions around the x-axis and y-axis, defining the radius function and limits of integration in each case to set up the integral
Calc 7.2b
The document provides instructions for calculating the volume of two different solids by revolving regions about axes and integrating with respect to the variable of integration.
For the first solid, the region is bounded by y = x^2 + 1, y = 0, x = 0, and x = 1 when revolved about the y-axis. The region is split at y = 1 and the volume is found using disks and washers, integrating from y = 0 to y = 1 and y = 1 to y = 2.
For the second solid, the region is bounded by y = 1/2x and y = -1/2x + 1 when revolved about the x-axis. The base
Calc 7.1a
This document discusses calculating the area between two curves using integration. It explains that the area between curves f(x) and g(x) from x=a to x=b can be found by integrating f(x)-g(x) from a to b. It provides examples of finding the area between curves, including when the curves intersect multiple times, requiring splitting the region into separate intervals. The key steps are sketching the region, identifying intersection points, determining which curve is above the other in each interval, and setting up the correct integral expressions with the proper limits of integration. Representative rectangles are used to visualize the accumulation of area, whether vertical or horizontal depending on the problem.
Calc 5.8b
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7.4 A arc length
- 2. An arc can be approximated by adding up small line segments, using the distance formula 2 2 2 1 2 1( ) ( )d x x y y A rectifiable curve is one that has finite arc length. It is also continuous, and its graph is a smooth curve. By letting the points on the curve get closer and closer, you end up with more and more line segments.
- 3. 2 2 1 n i i i s x y 2 2 2 2 1 n i i i i i y x x x Factor out and take square root, and you get 2 ix 2 1 1 ( ) n i i i i y x x Which results in the definition of arc length: Let y = f(x) represent a smooth curve on the interval [a, b]. 2 1 [ ] b a s f x dx
- 4. Similarly, for a smooth curve given by x = g(y), the arc length of g between c and d is 2 1 [ ] d c s f y dy
- 5. Examples: Find the arc length of the graph of 3 1 6 2 x y x in the interval [0.5, 2]
- 6. Example: Find the arc length of the graph of 3 2 1y x from x = 0 to x = 8, (Hint: solve for x, change over to y- limits