A 6-inch-tall plastic cup is shaped like a surface obtained by rotating a line segment in the first quadrant about the x-axis. Given that the radius of the base of the cup is 1 inch, the radius of the top of the cup is 2 inches, and the cup is filled to the brim with water, use integration to approximate the volume of the water in the cup.

Q: Find the volume of a right circular cone with height 18 and base radius 9 using integration. (Disk…

A: Solution:- by using disk washer formula

Q: Find the volume of the solid with a circular base of radius 5 whose cross sections perpendicular to…

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Q: The region in the first quadrant that is bounded above by the curve y= 1/x, on the left by the line…

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Q: Find the volume of a pyramid with height 24 and rectangular base with dimensions 5 and 7 using…

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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: calculate (by integration, showing all working) the distance from the nose of the jar to the centre…

A: Given figure: Given, z2=14yz=12y

Q: A water tank is in the shape of an inverted square pyramid. The pyramid is 10 ft tall and 24 ft wide…

A: Not putting in the limits as you asked..  And, when we say top is at origin, i.e. at x = 0, then…

Q: By iterated integration, find the area bounded by the parabola y2 = 4ax and its latus rectum.

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Q: Find the volume of the solid whose base is in the first quadrant bounded by y = √√7-7x, the x-axis,…

A: To find- Find the volume of the solid whose base is in the first quadrant bounded by y = 7 - 7x, the…

Q: (1 point) The volume of the solid obtained by rotating the region enclosed by y = a², y = 2x, about…

A: y=x2y=2x about line x=2 for point of intersection x2=2xx2-2x=0x(x-2)=0x=0,2

Q: The area bounded by y = 2x² - x³, and y=0 is revolved about x=0. Find the volume of the solid…

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Q: Use integration by parts (Tabular) to evaluate:

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Q: A cylindrical tank of height 5 feet and radius 0.5 foot is filled with oil that weighs 55 pounds per…

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Q: By iterated integration, find the area bounded by the parabola y2 = 4ax and its latus rectum, and…

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Q: The edge of a cube was found to be 107 cm with a possible error in measurement of 0.03 cm. Use…

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Q: find the centroid of volume if the volum is equal to 7/15 π

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Q: Find the volume of the solid whose base is the semicircle y = /25 – x² where –5 < x < 5 and the…

A: Given the base of the solidy=25-x2 , y&gt;0The intersection of the base with the x-axis :Put y =…

Q: Use the 4-segments multiple Trapezoidal rule to estimate the integral L VI+ z*dz

A: Given that  The integral is  To find the integral by using trapezoidal rule

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

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Q: By performing a 3-dimensional integral, calculate the volume of a hemispherical shell

A: Volume of hemisphere is

Q: A ball of radius 16 has a round hole of radius 8 drilled through its center. Find the volume of the…

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Q: Set up an integral to represent the total mass of a 2×5 rectangular sheet, whose density per unit…

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Q: A tank that has a shape of an inverted circular cone with height 15 meters and base radius 6 meters…

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Q: If a force of 128 lbs is required to stretch a 10-ft spring by 4 ft. How much work is done by…

A: Here we have to use the concept of integral calculus

Q: Find the volume of the solid whose base is in the first and second quadrant(s) bounded by y = 32 –…

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Q: Suppose that a semicircular wire has the equation y= V25- x and that its mass density is 8 = b- y.…

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Q: If the vase is filled with water with density 1000 kg/m², set up an integral representing how much…

A: Part(a):   The volume of the vase:   First find the area of the vase is :

Q: Find the volume of a pyramid with height 29 and rectangular base with dimensions 6 and 9 using…

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Q: Find the area of the surface formed by revolving y = x³ on 0 < x<1 around the x-axis. Show all steps…

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Q: The base of a solid is the region in the first quadrant bounded by the line y = -1/2x + 2 and the…

A: At any point x calculate the area of cross section and integrate it over the interval.

Q: The dimensions of a closed rectangular box are measured as 97 cm, 60 cm, and 30 cm, respectively,…

A: Let the dimensions of the box be l, w and h (for length, width and height).Surface area of the box…

Q: 3. Find the volume of a sphere with radius r by integration.

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Q: Find the surface area of the portion of the half ball with radius a which cuts off by the cylinder…

A: The equation of a ball with radius a is x2+y2+z2=a2. The surface area of the function z=fx,y over…

Q: 2. Find the volume of a cylinder with a radius of r and height h by integration.

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Q: The dimensions of a closed rectangular box are measured as 88 cm, 53 cm, and 32 cm, respectively,…

A: Given: The dimensions of closed rectangular box is : =88cm×53cm×32cm

Q: Find the dimensions of the right circular cone of the greatest lateral area that can be inscribed in…

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Q: Find the volume of the solid with a circular base of radius 4 and the cross sections perpendicular…

A: Given, the radius is 4 so we can conclude x2+y2=4 (equation of circle) =y2=4-x2 y=4-x2 one side of…

Q: Suppose a cylindrical tank with a height 8 m and radius 4m is half-full of water. Assume the density…

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Q: 2. By integration, show that the area of a circle of radius r is A = rr?

A: According to our guidelines, we can answer only one question or three subparts of a question. Other…

Q: The dimensions of a closed rectangular box are measured as 74 cm, 51 cm, and 50 cm, respectively,…

A: For a function y=fx, ∂y,∂x,dy and dx represents partial derivative of y and x and then differential…

Q: use the table methud from integration by parts to find S(4x2-5x+6e clx

A: we have to find the integral by using integration by parts

Q: The axes of two circular cylinders of radius a intersect at right angles. Show that the volume…

A: 8∫0ady∫0a2-y2dx∫0a2-y2dz = 8∫0ady∫0a2-y2dx (a2-y2)8∫0ady∫0a2-y2dx∫0a2-y2dz  =8∫0ady…

Q: ** By using horizontal element, the definite integral representing the area of the reg pounded by x…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: Sre* dr

A:

Q: Find the volume when the area in y = 2x^2 − 8x + 12 and y = −2x + 12 is 9 sq units and revolves…

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Q: Make an appropriate substitution and then use integration by parts to evaluate the integral.…

A: Take the constant out of the integral sign,

Q: A hole is drilled through the center of a ball of radius r, leaving a solid with a hollow…

A: Given  that the center of a ball of radius r;  and  the  height of the cylinder be…

Q: Find area of circle whose eq is (x-3)^2 - (y-3)^2=4 using integration

A: Note: The correct equation of circle will be x-32+y-32=4.  The radius of the circle is 2 and center…

Q: A pool in the shape of a hemisphere with a radius of 7 feet is filled with 6 feet of water.  Letting…

A: Letting (0,0) be the point on which the tank touches the ground, the center of the sphere would be…

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.