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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 12T

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Question

Solid S is bounded by the given surfaces. (1) Sketch S and label it with its
boundaries. (2) Give the inequalities that define S in polar coordinates. (3) Find
the volume of S using double integral in polar coordinates.
2 = √√√√x² + y², 2:
x² + y²,
z=0, x= √√1-y², x = 0, y ≤ 0.
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Let the solid S be bounded by an elliptic paraboloid z = 9-x2-y2 and planes x = 0, y = x & z = 0. Set up a double integral in rectangular coordinates that will give the value of the volume of solid S. And, this time, using polar coordinates, use double integrals to evaluate the volume of solid S.
Consider a curve represented by x2 + y2 = 4x.(a) Find the polar equation of f.(b) Set up the integral to find the area inside the curve f and outside r = 2.
consider the function in polar coordinates f(r, θ) = (cos(2θ) − r) sin(πr) on the region inside the circle r = 1. Set up the double integral in polarcoordinates to find the volume of the region below f(r, θ) and above the xy-plane.
Consider I =E∫x32dx1dx2dx3 , where x12 + x22 ≤ x32 , 0 ≤ x3 ≤ 1 ,that is, E is the solid bounded by the cone x12 + x22 = x32 , the plane x3=0, and the plane x3=1 (a)Sketch the solid E' in spherical coordinates that will correspond to E. (b)In the application of Fubini's theorem to the E′ solution of the question (a) above, with the orderof integration ∫∫∫ρ4cos2(ϕ)sen(ϕ)dρdϕdθ explain, illustrating in the sketch of solid E', how we find the lower and upper ends of the iterated integrals (c)Use spherical coordinates to calculate I. Some data may be needed (see image below): In the formulas below E, S and l always denote a solid, a surface, and a line, respectively. While n(x) denotes the normal unitary exterior of S in x, and T(x) denotes the unitary tangent of l in x.

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