A right circular cone (with the vertex at the bottom) has a height of 6m, and a radius at the base of 3m. The cone is full of water. Recall that the density of water is p = 1000 kg/m³ and gravity is g = 9.8 m/s². Set up an integral which computes the amount of work performed in pumping water over the top of the cone until 2m of water remain in the cone. Show a diagram as part of your work. Identify on your diagram: the y-coordinate at the top of the cone, the y- coordinate at the bottom of the cone, and the volume and thickness of an arbitrary "slice" of water.

Q: Set up the integral to find the surface area of the volume generated by the curve y=1/2 (x −3)^2…

A: Given, a=0,b=2  y=1/2 (x −3)^2 +1/2 Therefore, (2y-1)/2 = 1/2 (x −3)^2 2y-1= (x-3)² Taking square…

Q: - S (4 — х 2y) dxdy

A:

Q: A cylindrical tank has a radius of 6 ft and a height of 12 ft. Set up an integral to determine how…

A: (a) The height of the cylindrical tank is 12ft and its radius is 6ft. We need to find the work done…

Q: Set up the integrals that would find the volume of the indicated region. The region common to the…

A: We will find the limits of x, y and z

Q: A water trough has a semicircular cross section with a radius of R = 2 meters and a length of L = 3…

A: Consider the 2 dimensional view of the semi circular tank Let trough is located at 0,0First we…

Q: Set up an integral for the volume of the solid obtained by rotating the region bounded by the given…

A: Hey, since there are multiple subparts posted, we will answer the first question. If you want any…

Q: Q1: set up (but don't evaluate) an iterated triple integral for the volume of the solid enclosed…

A:

Q: Interpret TV? + y? dydr as a volume and calculate the integral without integrating.

A: We have to interpret the given integral as volume.

Q: Find the mass M and center of mass x of the linear wire covering the given interval and having the…

A:

Q: The volume generated by revolving the area bounded by the curve y = x² and y = 4 around the line y =…

A: We can use the formula π∫abR(x)2dx to evaluate the volume.

Q: Determine the volume of the solid of revolution formed by rotating the region bounded by the…

A:

Q: Plane न्य to

A:

Q: 5. Set up the integral to find the volume of the solid whose horizontal base is a circle of radius 2…

A:

Q: Set up a sum of definite integral(s) that is equal to the volume of the solid whose base is R and…

A:

Q: Then, compute the surface area of the solid that is formed by rotating y = In(x) on [0, n] about the…

A:

Q: Find the volume of the solid that results when the region enclosed by y= x² and y = 12 - x is…

A:

Q: Use a double integral to find the volume of the indicated solid. X+ y +z = 6 731

A: we have to find the volume using double integral.

Q: Find the volume generated if the area bounded by the curve x² = y – 2 is revolved about the x-axis.

A:

Q: If we want to find the area between the curves x = −y 2 + 1 and x = y 2 − 1, shall we integrate with…

A: Given curves are x=-y2+1 and x=y2-1

Q: A solid is constructed between the graphs of y = f(x) and y = g(x) and the lines a = a and a = b so…

A: Let a triangular slice with thickness dx at a distance of x from the origin. The height of the…

Q: A hole of radius is bored through a cylinder of radius R>rat right angles to the axis of the…

A: A hole of radius is bored through a cylinder of radius R>rat right angles to the axis of the…

Q: 10 3. sec x dx

A:

Q: Show that if a solid figure is revolved about the x-axis, its volume of revolution is given by V= a…

A: To show that if a solid figure is revolved around the x-axis, its volume of revolution is given by…

Q: A tank in the shape shown below is filled with water. Set up the integral that would calculate the…

A: To determine the amount of work needed to pump the water.

Q: Find the volume of the area bounded by the line y = 3x + 4 and the curve y = x² if it is revolved…

A:

Q: 20. V9 – x² dx

A:

Q: Sketch the region whose area is given by the definite integral. The use a geometri formula to…

A: Explanation: Given that,                                        We will graph the function…

Q: A water tank has the shape of an upright cylinder with a base radius r = 2 meters and a height h = 5…

A: Work done bringing up a mass m to height h is mgh Volume of a cylinder is pir^2h

Q: Use a double integral to find the volume of the indicated soli 40 z 30 z = 25 – xy 20

A: Given: z=25-xy y=x    and   y=5

Q: Find the surface area generated by revolving about the y-axis that part of the curve y = In x lying…

A: Solution We have to find the Surface area generated by revolving about the y-axis the part of curve…

Q: A water tank has the shape of an upright cylinder with a base radius r = 2 meters and a height h = 5…

A: Given:for cylinder radius of baser=2 meters.              heighth=5 meterswater level filled in…

Q: A rectangular aquarium having a length of 3 meters, a width of 2 meters, and a height of 2 meters is…

A: Consider the provided question, We need to find the work required to the pump. consider the below…

Q: Find the volume generated when the area bounded by the two curves of x^2 = 4y and y^2 = 4x is…

A: We will find out volume of solid of revolution.

Q: Find the volume of: the area bounded by the curve y2 = 9x and the line x = 4 revolved about x = 4.

A:

Q: Consider a cylindrical tank that has a height of 12 m and a radius of 5 m. Recall that if the tank…

A: Assume the height from the bottom to the slice is y and the remaining height from slice to above the…

Q: Find the volume of the solid generated when the region bounded by y = = 0, x(9 – x x = 7, and x = 8…

A: The given curve is,

Q: Approximate the value of the gven Integral by ase of the Trapezoidal Rule, using the giveu Value of…

A:

Q: A solid right circular cone of height h with a 90° vertex angle has density at point P numerically…

A: Given that a solid right circular cone of height h with a 90° vertex angle has density a point P on…

Q: The integral for the volume of the solid obtatned by rotating the regton bounded by the curves y =…

A: Volume is 2πe where e is exponential.

Q: Find the volume of the solid that results when the region enclosed by y= \x, y=0, and x =9 is…

A:

Q: 2. Find the volume of the solid that results when the region enclosed by y= x? and y = 12 –x is…

A:

Q: Set up the integrals to find the volume generated by revolving the region bounded by y = 4 – 2x and…

A: here is the solution

Q: Find the volume generated when the area bounded by the curve y2=x , the line x=4 and the x-axis is…

A: Volume bounded by the curves, about y-axis.using Disk method

Q: The area bound by the curve y^2 = 12x and the line x = 3 is revolved about the line x = 3. What is…

A:

Q: (2,2,0)

A:

Q: Evaluate the integral by reversing the order of integration. So Soy e dædy =

A: Given: ∫01∫8y8ex2dxdy

Q: Determine the exact value of the definite integral by interpreting it as a measure of the area of a…

A: We have to integrate

Q: What is the exact volume of a solid that is formed by rotating the region enclosed by the functions…

A:

Q: Use a double integral to find the volume of the indicated nolid. 12 z- 12-X-y 11 10 4. 6. y 6 y-x

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.