3. Suppose that a ring created by drilling a hole with radius r through the center of a sphere of radius R. The volume of the ring is twice the volume obtained by rotating the area above x-axis and below the curve y = vR? – r? between x = r and x = R about the y-axis. Refer to Figure 1. y= VR? – x Figure 1. By using cylindrical shells method, compute the volume of the ring and express the answer in terms of h. ( Hint: Pythagorean Theorem, R² - r² = (;h)' | Hint: S(27 radius - high)dx ] b. Given a parabola function of y = x? and a function of y = x*. State the intersection values between the given functions.
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.
Using Handwriting, answer (3.a)
![3. а.
Suppose that a ring created by drilling a hole with radius r through the center
of a sphere of radius R. The volume of the ring is twice the volume obtained
by rotating the area above x-axis and below the curve y = VR? – r²
between x = r and x = R about the y-axs. Refer to Figure 1.
y=VR?.
Figure 1.
By using cylindrical shells method, compute the volume of the ring and
express the answer in terms of h.
( Hint: Pythagorean Theorem, R² – r² = (;h)´
( Hint: S(2m - radius · high)dx ]
Given a parabola function of y = x? and a function of y = x*
b.
i.
State the intersection values between the given functions.
ii. Find the area of the region.
i.
Sketch the region enclosed based on answer in 3. (a)(i) and 3. (a)(i).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda8d7128-afc7-4b3e-85fb-fe3ec9ab8f04%2Fe542a508-0e3c-4867-9236-003704cbda76%2Fbix94r_processed.jpeg&w=3840&q=75)
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