Let S be the tetrahedron in TR 3 with vertices at the vectors 0, e1, e2, and ez, and let S' be the tetrahedron with vertices at vectors 0, v1, V2, and v3. See the figure. X3 X3 S' x2 a. Describe a linear transformation that maps S onto S’. b. Find a formula for the volume of the tetrahedron S' using the fact that {volume of S} = (1/3){area of base} · {height}

Q: Let C be the positively oriented square with vertices (0,0), (1,0), (1, 1), (0, 1). Use Green's…

A: Let's solve by using green theorem.

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), E (0,4,0), and (0,0,2)

A: The vertices of the tetrahedron are (0,0,0),(1,0,0),(0,4,0),(0,0,2).  x varies from x:0→1y varies…

Q: In AABC, X is the centroid. If CW = 15 , find XW. %3D

A: Given,           In △ABC, X is the centroid and CW¯=15

Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

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Q: Let T be the tetrahedron with vertices (2, 0, 0), (0, 3, 0), (0, 0, 5) and the origin. Write (a, y,…

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Q: Find the y coordinate of the centroid of the region shaded in the following diagram. 9 in. -27 in.

A: Here the figure has drawn for our convenience.  To find the centroid, the rectangle indicated in x…

Q: C is the hexagon in the xy-plane with vertices (5, −5), (5, 5), (0, 10), (−5, 5), (−5, −5), and (0,…

A: Given : Vertices : (5,-5) , (5,5), (0,10), (-5,5), (-5,-5) and (0,-10) Integral given :…

Q: Compute the distance d from y to the plane in R' spanned by u1 and uz.

A: We have to find the distance d from y to the plane in R^3.

Q: EXAMPLE 2 Find the area of the part of the paraboloid z = x + y' that lies under the plane

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Q: Find the centroid (x, y) of the triangle with vertices at (0, 0), (10, 0), and (0, 2). 18 15 X=

A: Consider a triangle with vertices at (0,0),(10,0) and (0,2)

Q: Z is the centroid of AUVW. What is UY if UZ = 8? O A. 4 о с. 6

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Q: Let H be the segment of the hyperbola x-y°= | Connecting (1,0) to (2, J3). Evaluate the line,…

A: Given  H be the segment of hyperbola x2-y2=1 connecting the points (1,0) to (2,3)  we have to…

Q: ind the volume of the solid that lies under the hyperbolic paraboloid z = 3y2 - x2 + 4 and above the…

A: Using the double integral formula.

Q: Let T be the tetrahedron with vertices (2, 0, 0), (0, 2, 0), (0, 0, 3) and the origin. Write I.f(r,…

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Q: Determine the distance h for which the centroid of the shaded area is as far above line BB' as…

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Q: 2. Let W be the union of the first and third quadrants in the xy- plane. That is, let W = xy 0 a. If…

A: According to our company gardening can answer only one question. If a specific questions are asked…

Q: In ANPQ, U is the centroid. If NS=15 find US. R S U

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Q: Find the image of the semi-infinite strip in the z – plane under the Transformations given by @) to…

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Q: 4. Locate the centroid (x, y)of the following composite areas: -3 ft 3 ft I 1.5 ft 1 ft

A: Consider the given composite area and divide then into four areas as shown in diagram below,

Q: Let S be the passage of R ^ 2 defined by the parabola with equation y = 18-x ^ 2 and by the line…

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Q: Use the liner transformation T: (u, v) → (x, y) given by 1 1 T(u, v) = (-u + v)i + ÷(u+ 3v)j 4 4 to…

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Q: Evaluate the triple integral. 6xyz dV, where T is the solid tetrahedron with vertices (0, 0,0), (1,…

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Q: Let D be the triangular region in the ry-plane with vertices (0,0), (, 0), and (1, 1). Let M be the…

A: Consider the given triangular region D in the xy-plane. The vertices are 0,0,1,0,1,1. Now, draw the…

Q: Find the coordinates of the centroid (to the nearest 0.01 in.) of the uniform flat-plate machine…

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Q: 2. In A ABC, X is the centroid. A A If CW = 15, find cx and XW. W, B. If BX = 8, find BY and XY.

A: we know that  centroid is located 23 of the distance from the vertex along the segment that connects…

Q: Let T be the tetrahedron with vertices (3, 0, 0), (0, 3, 0), (0, 0, 5) and the origin. Write z) dV…

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Q: 4. Given an example of a change of coordinates z= r(s,t), y = y(s,t) where every region in the st…

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Q: 4. A 5m x 5cm is cut from a corner of 20cm x 30cm. cardboard, find the centroid from the longest…

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Q: Evaluate || F-n dS where S is the cylinder (including the disk on the bottom) x² + y² =1 for 0<z<1…

A: First, we plot the given surfaces.Cylinder : x2+y2=1; 0≤z≤1Paraboloid : 3x2+3y2+z = 4; 1≤z≤4.

Q: Find the centroid of the given region. (A) (5,15) (B) (3/22,45/61) (C) (5/14,38/35) (8/23,76/111)…

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Q: Use the given transformation to evaluate the integral. 15) x = 6u, y = 9v, z = 2w; x2 y2 z2 dx dy…

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Q: 3. Let H = 2xi – 3xy j+xz² k. Calculate ſ, Ĥ •d where S is the surface of the cube with corners at…

A: Given- Let H→ = 2xi→ - 3xyj→ + xz2k→ To find- Calculate ∫SH→·dA→ where S is the surface of the cube…

Q: Evaluate the triple integral.  (x2 + y2) dx dy dz; W is the pyramid with top vertex at (0, 0, 2)…

A: The given triple integral is,

Q: Find the coordinates of the centroid (to the nearest 0.01 in.) of the uniform flat-plate machine…

A: The solution is given below

Q: Find the centroid of the polyhedron with four triangular faces in the first octant that is bounded…

A: The region is bounded by the three coordinate planes and the plane x+y+z=1. Thus, the region can be…

Q: 3) Let D be the region enclosed by the two paraboloids z = 4 + x² + y² and z = 10-x² - y². Then the…

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Q: Find the centroid of the solid bounded by the parabolic cylinder 1- y? and the three planes given…

A: To find the centroid of the solid bounded by the parabolic cylinder: The region is defined as…

Q: Evaluate the circulation of Gayi + zj+5yk around a square of side 9, centered at the origin, lying…

A: Solution :

Q: Q1- Determine the coordinates of the centroid of the area.: (1, 1) y= x,

A: The given problem is to find the centroid of the given shaded area of the region formed by given…

Q: Consider the transformation x = r cos Oy =r sin = zfrom cylinderical to rectangular coordinates,…

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Q: Let T be the tetrahedron with vertices (5, 0, 0), (0, 2, 0), (0, 0, 5) and the origin. Write II|…

A: Given T is a tetrahedron with vertices (5,0,0) , (0,2,0), (0,0,5) and  (0,0,0)…

Q: Evaluate the integral // (10 · x + 9 · y)dA, where R is the parallelogram with: R vertices (-3,6),…

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Q: Let B be the region in the first quadrant of the ay-plane bounded by the lines r+ y = 1, x + y = 2,…

A: Consider the given information.

Q: Find the coordinates of the centroid (to the nearest 0.01 in.) of the uniform flat-plate machine…

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Q: Q1- Determine the coordinates of the centroid of the area.: (1, 1)

A: We need to find the x coordinate and y coordinate of the shaded region which is bounded by two…

Q: Find a Möbius transformation which maps the region outside the unit circle onto the left- half…

A: We have to find a Mo¨bius transformation which maps the region outside the unit circle onto the…

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (5,0,0), (0,6,0), and (0,0,3)

A: This question is related to triple integral, We will solve it using given information.

Q: Find the centroid (, j) of the triangle with vertices at (0, 0), (8, 0), and (0, 9). x=

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Q: Use the given transformation to evaluate the integral. J (8x + 8y) dA, where R is the parallelogram…

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