A ring is formed by rotating the shaded area 360° about y-y axes. Determine (1) its surface area and (2) its volume. y 50 mm 40 mm 20 mm y

Q: Use polar coordinates to find the volume of a sphere of radius a.

A:

Q: a) Find the volume and the lateral surface area of the solid formed by rotating this func- tion…

A:

Q: Draw a cylinder and sketch a horizontal and vertical cross section on the figure.

A:

Q: Find the volume of the pyramid with base in the plane z = -1 and sides formed by the three planes y…

A:

Q: A frustum of a cone has base radii 2 and 5 and height h=4 h What is the surface area of the frustum…

A:

Q: I In Exercises 10–11, find the surface area. 10. A cylinder of radius a and length L.

A: Solution is given as below:

Q: The center of each two spheres having equal radius of 2 m. lies on the surface of the other sphere.…

A: Here in this question we have to find out the volume common to the two spheres of equal radius. The…

Q: Using spherical coordinates, find the volume of the sphere with radius a

A:

Q: find The volume of Ihe solid generated by revolving each vegion enclosed by he triangle wilh…

A:

Q: Use cylindrical coordinates to find the volume of the region it

A: Given, the region in the first octant bounded by the cylinder r = 5 and the plane z=x.

Q: Calculate the surface area of a sphere of radius a.

A:

Q: Find the volume of the torus generated by revolving the circle (x - 2)^2 + y^2 = 1 about the y-axis.

A: Given: (x-2)2+y2=1

Q: Find the volume of the torus formed by revolving the circle x2 + y2 = a2 about the line x = b(b >…

A:

Q: A three dimensional solid is created by rotating a circle about one of its diameters. What is the…

A:

Q: Find the volume of the torus generated by revolving the circle (x−5)2+y2=4 about the​ y-axis. What…

A: Here, equation of circle is: (x-5)2+y2=4 So, the centre of the circle is (5,0) and radius will be 2.…

Q: Use the Theorem of Pappus to find the exact volume of a general cone created by rotating a triangle…

A:

Q: Find the volume of liquid needed to fill a sphere of radius R to height . R y

A:

Q: The circle x2 + y2 = a2 is rotated about the x-axis to generate a sphere. Find its volume.

A: we have been given that the circle x2 + y2 = a2 is rotated about the x-axis to generate a sphere. we…

Q: A yo-yo is made from a sphere by removing a slice from its center, as in Figure 6.35. Suppose the…

A:

Q: a pluke is composed of hwo semi - cireles, y= = vi- x? and a y> Ju-x?. find the center of mass of…

A: Given the semicircles are  y=1-x2y=4-x2 For the first semicircle, y=1-x2y2=1-x2x2+y2=1 So the radius…

Q: Determine the volume generated by revolving the plane figures about y axis. 4m 3m 1m 2m 4m 2m

A:

Q: A body T with the parameters  (1/2)(sqrt(x^2)+(y^2))≤z≤sqrt(16-(x^2)-(y^2) Use the sphere…

A: The given problem is to find the volume of the body formed by the cone and spherical region, we have…

Q: Use cylindrical or spherical coordinates, whichever seems more appropriate. Find the volume of the…

A: Given:A sphere of radius 10 by two planes that intersect along a diameter at an angle of π/2.

Q: 8. 9. 2- -5 5. 10 -2-

A:

Q: Solve for the radius of a right circular cone of maximum volume which can be inscribed in a sphere…

A:

Q: Use cylindrical or spherical coordinates, whichever seems more appropriate. Find the volume of the…

A:

Q: Find the circular cylinder of largest lateral area which can be inscribed in a sphere of radius 4…

A:

Q: Prove mathematically that a cylinder with hemispherical ends that has the same volume as a sphere…

A: Let r be the radius of the hemispherical end. Let h be the height of the cylinder. Let r' be the…

Q: Solve for the radius of a right circular cone of maximum volume which can be inscribed in a sphere…

A:

Q: Find the area of the adjacent surface of a triangular prism whose base is a triangle with sides a =…

A: Area is the measure of space occupied by a flat surface. It is measured in terms of the number of…

Q: A pyramid that has a square base with length 12 and height of pyramid is 8 and the lateral edge is…

A: For a square pyramid, we have base b = 12, height h = 8 and the lateral edge e = 11.7.

Q: Find the center of mass for a quarter of a regular circular plate of radius (b) b

A:

Q: 41. Find the volume of the solid which is the intersection of the two cylin- ders in Figure 9. Hint:…

A: Given: A solid with a circular base of radius a. Each plane perpendicular to the diameter AB…

Q: A torus is generated by rotating the circle x2 + (y – R)² = r² about the x-axis. Find the volume…

A:

Q: Find the volume V generated by rotated the

A: 32.  Given the curve                           f(x)=3xln3  over 1, 3

Q: Find the volume of the torus obtained by revolving the circle x^2 + y^2 = 4 about the line x = 3.

A: Find the volume of the torus obtained by revolving the circle x^2 + y^2 = 4 about the line x = 3.

Q: The vertices of a triangle are at (0,0), (6,6), and (9,3) . What is the volume, in cubic units, of…

A: The figure by hand is show below of the given question:

Q: The radii of the internal and external surface of a metallic spherical shell are 3 cm and 5 cm…

A: First, finding volume of shell and then volume of cylinder

Q: Find the volume of the washer that is obtained when the shaded area is rotated about the x-axis. ry…

A: Given that y=x+2,y=2 The formula for washer method is, V=π∫abf(x)2-g(x)2dx Here in the problem,…

Q: The volume of the sphere with the radius 7 is = %3D cubic units

A: Consider the given: It is given that:Radius,r = 7Volume of sphere = 43πr3

Q: A sphere with radius r is inscribed in a cylinder. Find the volumeof the cylinder in terms of r.

A: let a right circular cylinder of radius 'R' and height 'H' is inscribed in the sphere of the given…

Q: A regular hexagonal pyramid with base edge 6 and height 8 is inscribedin a cone. Find the lateral…

A: Pyramid: A regular hexagonal pyramid with base edge 6 and height 8 is inscribed in a cone.  For cone…

Q: What is the difference in volume between a sphere with radius rr and a sphere with radius 0.3r0.3r?

A: Given query is to find the difference in the volume.

Q: 6. Find the volume of the torus generated by revolving the circle x² + y² = 4 about the line .x = 3.

A:

Q: Use polar coordinates to calculate the volume of a sphere of radius R.

A: This question is based on application of polar coordinates integration.

Q: Find the volume of spherical segment of one base and altitude h cut from a  sphere of radius a.

A:

Q: 4. Use Maple to draw the torus whose equation in spherical coordinates is given by p= sin(ø), and…

A: Given Use Maple to draw the torus whose equation in spherical coordinates is given by ρ=sinϕ.

Q: (b) The volume generated by i. rotating the area enclosed between y = 1- 2² and y = 0 by the y axis.

A: Given that the area is between curves y=1-x2 and y=0. Rotated about y-axis.

Q: Find the volume generated by y = x^2, x=0 and y = 4 rotated about the y axis. with drawing please.

A:

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.