a) Show that the volume enclosed when revolving the curve y = f(x) - where f: [a, b] → [0, 0) - about the x-axis in three-dimensional x-y-z space is given by %3D V = T a

Q: 10. Show that the volume inside the ellipsoid ++=1 is 4abe (Hint: Use the change of variables x =au,…

A: We have to find volume of ellipsoid: Concept:

Q: In Exercises 29-34, find the volume of the solid obtained by rotating region A in Figure 13 about…

A: From the given information, the upper curve is y=6 and the lower curve is y=x2+2, The axis of the…

Q: B-Find the center of mass of the region bounded above by the half circle y = √a²-x² and below by the…

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Q: tan(y), y 4. Find the volume of the solid generated by revolving the region bounded by r = and r =1…

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

A: Given:z=3y2-x2+2 above the rectangle R=-1,1×1,5Formula:V=∬R z dx dy

Q: (b) Find the volume under the paraboloid z-x+y above the triangle enclosed by the lines y=x,x-0 and…

A:

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

A: The volume of the solid that lies under hyperbolic paraboloid is:

Q: Q.6 Find the volume of the solid generated by revolving the loop of the curve y =x(2-x) about the x-…

A:

Q: (3+ In x)² Find the volume of the solid when the curve y=- is rotated 360° about the x-axis between…

A: Here, the function is y=f(x)=(3+ln x)32x12. The volume obtained by rotation is given by…

Q: Given f(x)=4x+1 and g(x)= x²+x-3: Use the disk method to determine the volume of solid obtained by…

A:

Q: a) A laminar with density &(x, y) = lies in the first quadrant is bounded y3 by two circles a2 + y²…

A: Given that, the density function is: δ(x,y)=xy3 Now, Ix=∫∫x δ(x,y) dy dx   =∫∫xxy3dy dx   =∫∫x2y3dy…

Q: Let f(x, y) = 4x²y. Find the volume of the solid underneath f(x,y) and above the half-disk a² + y² <…

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Q: 3 27 Find the center of mass of a thin plate of constant density & covering the region bounded by…

A: Center of mass of a body is a point at which it can be considered that entire mass of body is…

Q: 5. Find the volume of liquid needed to fill a sphere of radius R to height h (Figure 18). R h…

A:

Q: Find the volume of the solid generated by revolving the curve y - 1* – 1 = 36 between z =0 and z=1…

A: Given that  Curve y^2-x^8-x^2=36 From x=0 to x=1  Find the volume of the solid

Q: In Exercises 27 and 28, sketch the solid whose volume is given by the specified integral. 27. (9 –…

A:

Q: In Exercises 17–20, find the volume of the solid generated by revolv- ing the shaded region about…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Let f(x) = 25 - x2. A solid is generated by rotating the reagion bounded by f(x), the x-axis and the…

A: We first sketch the region using a graphing calculator  the shaded region is rotating about the y…

Q: A gasoline tank is an oblate spheroid generated by revolving the region bounded by the graph of…

A:

Q: Suppose that a thin metal wire in the shape of a semicircle occupies the curve c, where c is the top…

A: Given a thin metal wire in the shape of a semicircle occupies the curve C, where C is the top half…

Q: Ex: Find the center of mass of the solid whose density &z and bounded above by the plane z=y, below…

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Q: 7.3 5. Use the shell method to set up and evaluate the integral that gives the volume of the solid…

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Q: 1. Consider the spiral ramp r(u, v) = ( u cos v, u sin v, v ) with 0 < u < 3 and 0<v < 2n. Use a…

A: Consider the given information.

Q: At what points (x,y.z) in space are the tunctions continuous? a. h(x.y.z) = In (4z² - 4x – 2y³ - 5)…

A: This is a problem of multi variable calculus.

Q: FD is the region in the first octant between the spheres x2+y2+z²=1 and x2+y2+z²=16, then dV =

A: Converting into spherical coordinates

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: Find the mass of a spherical shell of radius R such that at each point (x, y, z) on the surface the…

A:

Q: A lamina occupies the region inside the circle x2 + y2 = 6y but outside the circle x2 + y2 = 9. Find…

A:

Q: |Find the volume of the solid below The solid S whose base is the region R enclosed by the curve y =…

A:

Q: 1. Find the surface area of the cone z = Jx2 + y2 that lies between the plane y = x and the cylinder…

A: Surface area in xy-plane

Q: Let R be the rectangle in the xy-plane with vertices (+1,±1). Find the volume of the solid lying…

A:

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: 5. Find the volume of the solid of revolution formed by rotating the finite region bounded by curves…

A: Given :Curve :y=x-1 y=x-12

Q: 1. Compute the average height of the cone z = a> 0. r² + y² above the disk x² + y² ≤ a² for

A:

Q: Q2:A) find the volume generated from rotation between curves : y = tan(x) y=1 and the interval (, )…

A: The volume of a solid obtained by rotating the region between the region y=f(x) and y=g(x) from x=a…

Q: Example. Find the volume of the solid above the xy-plane bounded by the elliptic paraboloid z = x² +…

A:

Q: (b) Let F = (-yx², xy²). Let D be the annulus 1 < a2 + y² < 4, and let C be its boundary. Evaluate

A: Value of the given integral is

Q: 1. Find the volume under the parabolic f(x,y) = x²+ 4y² and above the rectangle R :0 < x < 1,1 < y <…

A: Given  fx,y=x2+4y2

Q: Consider a thin rod oriented on the x- axis over the interval [0, x/2]. If the density of the rod is…

A:

Q: 1) Calculate a)be the volume of the solid below the paraboloid 2+x²+(y-2)² and above the rectangle…

A: In this question, we find the volume of the solid bounded by paraboloid and base is a rectangle. we…

Q: Define R as the region bounded by the graphs of f(x)=4x2 −4x+2, x=0, x=1, and the x-axis. Using the…

A: We have to find volume of solid of revolution about x axis .F(x) and graph  is given below :

Q: O Find the volume of solid in the first octant bounded by the co- ordinate planes, the planes x-3,…

A: We will find out the required volume of the solid in the first octant.

Q: A- Find the volume of the solid generated by revolving region bounded by the curve y(x-1) and lines…

A: As per the question we are given a solid of revolution which is obtained by rotating the 2d region…

Q: The volume of the solid generated by revolving the 2y curve x = 4, about the y-axis between y = 0…

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Q: Let D be the solid whose base lies on the xy plane and is bounded by the graphs of y = 2x and y = 3…

A: To find the point where the two functions intersect, equate the two.   Cross-sectional area of the…

Q: Show that the subtangent to y = x' at x = a is equal to a/3.

A: Definition used -                            Length of subtangent =ydxdy

Q: In Exercises 25-28, calculate the approximation to the volume of the solidobtained by rotating the…

A: Plot the curve y = cos x in the given interval. From the graph the upper curve is y = cos x ad the…

Q: Cvaluate AP G is a solid in the first octant bounded by the plane y+ z = 2 and the urface y = 1 -…

A: