Question 1.(a)paraboloids z=3+x²+y? and z=2+2x2+ 2y2.Find the intersection of the(b)gives the volume of the region bounded by theConstruct the triple integral whichparaboloids given above.(c)Evaluate triple integral obtainedin part (b) to find the volume.

Using cylindrical coordinates to find the volume

Answered: Question 1. (a) paraboloids z=3+x2+y?…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. (2 points) State the formula for the area of an equilateral triangle with side length a. 2. (13 points) Consider the solid whose base is the region bounded by the curves y = Vx, y = 0 and x = 4, and whose cross-sections perpendicular to the x-axis are equilateral triangles. (a) (8 points) Use the slicing method to write an integral representing the volume of this solid. (b) (5 points) Find the volume by evaluating the above integral.
The figure shows a vertical cylinder that is cut diagonally by a straight plane. Create a multiple integral that allows you to determine the volume that is inside the cylinder, above the diagonal plane and bounded superiorly by the plane z = 2. The equation of the cylinder is x^2 + y^2 =9 and the equation of the diagonal plane is z = -2/3y +2.
Vi). Set up an integral that represents the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. Do not evaluate the integrals. Please do both
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calculate the integral
2. (Calculator Active) Let R be the region bounded by the x-axis and the graphs of y = 1+ x² and y = 4 – 2x , as shown in the figure above. y = 4 – 2x (a) Find the area of R. y = 1+ x? b) Find the volume of the solid generated when R is revolved around the line y = -2. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a semicircle. Find the volume of the solid.

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Question 1. (a) paraboloids z=3+x2+y? and z=2+ 2x2+ 2y2. Find the intersection of the (b) gives the volume of the region bounded by the Construct the triple integral which paraboloids given above. (c) Evaluate triple integral obtained in part (b) to find the volume.