Let C be the arc of the curve y = f(x) between the points P(p, f(p)) and Q(g, f(g)) and let R be the region bounded by C, by the line y = mx + b (which lies entirely below C), and by the perpendiculars to the line from P and Q. y= f(x) R P C y = mx+b Au 1. what the area of R is - mx - b][1+mf'(x)] dx 1 + m e [Hint: This formula can be verified by subtracting areas, but it will be helpful throughout the project to derive it by first approximating the area using rectangles perpendicular to the line, as shown in the following figure. Use the figure to help express Au in terms of Ax.] tangent to C at (x;, f(x;)) y=mx+b Au Ax 2. Find the area of the region shown in the figure at the left. 3. Find a formula (similar to the one in Problem 1) for the volume of the solid obtained by rotating R about the line y = mx + b. 4. Find the volume of the solid obtained by rotating the region of Problem 2 about the line y = x - 2.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
see picture question 4.
![Let C be the arc of the curve y = f(x) between the points P(p, f(p)) and Q(g, f(g)) and let R
be the region bounded by C, by the line y = mx + b (which lies entirely below C), and by the
perpendiculars to the line from P and Q.
y= f(x)
R
P C
y = mx+b
Au
1. what the area of R is
- mx - b][1+mf'(x)] dx
1 + m e
[Hint: This formula can be verified by subtracting areas, but it will be helpful throughout the
project to derive it by first approximating the area using rectangles perpendicular to the line,
as shown in the following figure. Use the figure to help express Au in terms of Ax.]
tangent to C
at (x;, f(x;))
y=mx+b
Au
Ax
2. Find the area of the region shown in the figure at the left.
3. Find a formula (similar to the one in Problem 1) for the volume of the solid obtained by
rotating R about the line y = mx + b.
4. Find the volume of the solid obtained by rotating the region of Problem 2 about the
line y = x - 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabc5aa9d-951a-40e1-86a3-f092042c4046%2F216c96d5-38f9-4b0b-a626-ad0c64b07fa4%2F6i16hbk_reoriented.jpeg&w=3840&q=75)
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