Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 68. The solid inside the cylinder r = 2 cos θ , for 0 ≤ z ≤ 4 – x
Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 68. The solid inside the cylinder r = 2 cos θ , for 0 ≤ z ≤ 4 – x
Solution Summary: The author explains that the volume of the solid inside the cylinder is 3pi .
Miscellaneous volumesChoose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables.
68. The solid inside the cylinder r = 2 cos θ, for 0 ≤ z ≤ 4 – x
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
y = −x2 + 11x − 30, y = 0; about the x-axis
V=
The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x = (y − 5)2, x = 16; about y = 1
V=
A thick spherical shell occupies the region between two spheres of radii a and 2a, both centred on the origin. The shell is made of a material with density p = A(x2 + y2) z2, where A is a constant. Hence, or otherwise, find the mass of the shell by evaluating a suitable volume integral.You may find the substitution u = cosθ useful.
Find the volume of the solid generated by revolving the region bounded by y = 4 − x^2 , y = 3x , and x = 0 about the x axis.
Be sure to do all of the following:
Draw a sketch.
Draw a representative rectangle, this helps you determine the variable of integration and method.
State method used: disk, washer or shell.
The integral(s) you are using to find the volume.
Clearly work out the integration.
Leave answer in EXACT form. Do NOT give decimals.
Chapter 16 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY