(b) Prove the volume of a parallelepiped with a vertex at the origin and edges ü,v, and w is given by V = |ū • (7 × w) %3D

Q: Determine the volume if the area common with y²-3x+7y=8 and x²-4x+3=y² is revolved about the…

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Q: Find the volume of the solid that lies under the hyperbolic paraboloid = 3y² – x² + 2 and above the…

A: Integral

Q: 5. Find the volume of the tetrahedron bounded by the coordinate planes and the plane 3x + 6y + 4z -…

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Q: 2. Suppose that the triangle with vertices (-1,-1), (0, 1) and (1,–1) is revolved about the y –…

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Q: Determine the volume if the area common with y^2 - 3x + 7y = 8 and x^2 - 4x +3 = y^2 is revolved…

A: Integration of function under certain limits with respect to any of the axis gives the area under…

Q: Define R as the region bounded by the graphs of f(x)=x2 +2x+2, x=−2, x=0, and the x-axis. Using the…

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Q: Use the Theorem of Pappus to find the exact volume of a general cone c with vertices (0, 0), (3, 0),…

A: Step:-1 Pappu's Theorem:- According to this theorem, the Volume V of a solid of revolution generated…

Q: Find the volumes of the solids whose bases are bounded by the graphs of y = x + 3 and y = x² - 3,…

A: So points are ( -2 , 1 ) and ( 3, 6 )

Q: Find the volume of the tetrahedron bounded by the plane z=4-4x-2y and the coordinate planes. I

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Q: If the volume of the region bounded above by z = a2 − x2 − y2, below by the xy−plane, and lying…

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Q: Find the volume of the region E bounded by the paraboloids z = x² +₁² and z = 72 - 7x² - 7x².

A:

Q: Find the "volume" of the four-dimensional pyramid bounded by w +x+ y+z-1=0 and the coordinate planes…

A: The volume of the four dimensional pyramid is given by, V=∫v1 dv   =∫dzdydxdw. The boundaries are…

Q: 3. Find the volume of the tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x + %3D y + 2z =…

A: Will use the triple integration to find the volume of the tetrahedron which is bounded by the given…

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

A: We need to evaluate volume.

Q: Use the Theorem of Pappus to find the exact volume of a general cone created vertices (0, 0), (1,0),…

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Q: Find the volume of the figure with base bounded by the given curves such that slices perpendicular…

A: To find- Find the volume of the figure with base bounded by the curves such that slices…

Q: Find the volume enclosed by the cy linder (x-1) ty²=1 and the sphere x² +y²7z²=L4.

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Q: Reduce the surface integral ( S A.ds) of a cube of sides parallel to the three perpendicular axes,…

A: Here we can use divergence theorem to reduce the given surface integral. Divergence theorem states…

Q: The volume of the tetrahedron bounded by the coordinate planes and the plane x+y+z=4 is

A: Given

Q: Find the volume of the region bounded above by the elliptical paraboloid z= 6-2x - 2y and below by…

A:

Q: Determine the volume if the area common with y^2-3x+7y=8 and x^2-4x+3=y^2 is revolved about the…

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Q: Find the dimensions of the box with volume 1728 cm3 that has minimal surface area. (Let x, y, and z…

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Q: Find the volume of c such that the area of the region bounded by the parabolas y = 16x2 - c2 and y =…

A: Given parabolas are y1=16x2-c2y2=c2-16x2 The area between the given parabolas is 1283

Q: Suppose that a region R has area A and lies above the x-axis. When R is rotated about the x-axis, it…

A: Use the disc method to find the W in terms of V, k and A.

Q: If the region that lies inside the cardioid r = 1+ cose and outside the cirele r = 1 is the base of…

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Q: (d) Consider the parallelepiped P in R³ whose adjacent sides are (0, 4, 0), (3, 0, 0) and (-1, 1,…

A: Since the student has posted multiple questions and does not mention any specific question, so we…

Q: Use the Theorem of Pappus to find the volume of the given solid. The solid obtained by rotating the…

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Q: A cone with height h is inscribed in a larger cone withheight H so that its vertex is at the center…

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Q: The area bounded by the parabolas (x − 2)^2 − 9py=0 and y^2=9p(x−2) is to be revolved around x = 2.…

A: Suppose the region R is revolved about the line L, then the volume of the solid is calculated by the…

Q: Use the disk method to find the volume of the solid of revolution bounded by the x-axis and the…

A: We first graph the function using a graphing calculator

Q: If the region that lies inside the cardioid r = 1+ cos 0 and outside the cirele r = 1 is the base of…

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Q: If the area bounded by the x-axis, the lines x=5 and y=4, and the parabola y^2=-2x+4 is revolved…

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Q: Use the slicing method to find the volume of the solid of revolution bounded by the graphs of f(x) =…

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Q: Find the volume of the largest closed rectangular box in the first octant having three faces in the…

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Q: A cone with height h is inscribed in a larger cone withheight H  so that its vertex is at the center…

A: By similarity of triangles in the large cone and the small cone formed opposite to the cone with…

Q: (a) COMPUTE the dimensions of the rectangular box of maximum volume that can be x² + y? + z? = 4.…

A: Since you have posted multiple questions, we will solve only first question for you. To get…

Q: Chapter 6, Review Exercises, Question 011 Find the volume of the solid whose base is the region…

A: We have to find the volume of the solid whose base is bounded by the two curves y2=x and y1=1x.…

Q: The base of a solid is the triangular region bounded by the y-axis and the lines 3x + 2y = 6 and 3x…

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Q: Find the constant k > 0 such that the volume of the region lying inside the sphere x2 + y2 + z2 = a2…

A: Given that,k>0 is a constant such that the volume of the region lying inside thesphere…

Q: Find the volume of the solid that lies under the hyperbolic paraboloid z = 20 + x² - y? and above…

A: We have given z = 20 + x2 − y2 and R = [−2, 2] × [0, 4]

Q: Determine the volume if the area common with y? – 3x +7y = 8 and x² – 4x + 3 = y² is revolved about…

A: Answer is mentioned below

Q: The base of a solid is the region in the first quadrant bounded by the coordinate axes, and the line…

A: Volume of the solid

Q: Define R as the region bounded by the graphs of f(x)=x2 +4x+5, x=−3, x=0, and the x-axis. Using the…

A: Please see the diagram on the white board. The line in blue is the plot of the given curve f(x)=x2…

Q: Show that the volume of the region bonded by the cone z=/x +y and the paraboloid z x+y is a/6.

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Q: If the volume of the region bounded above by z = a²-x² - y2, below by the xy-plane, and lying…

A: Please see the attachment

Q: Determine the volume of the largest rect an gu lar box in the first oct ant with three faces in the…

A: Given: The volume of the largest rectangular box is to be determined based on the first octant with…

Q: Prove that if two solids have equal altitudes and all cross sections parallel to their bases and at…

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Q: The volume of the tetrahedron bounded by the plane * ++ = :1, {a, b, c > 0} and the coordinate a…

A: Volume of the figure will be given by integral - V = ∫∫∫dzdydxLet's find limits of z,y and x - Given…

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.