Problem 5
Two surfaces \(f_{1}(x, y)\) and \(f_{2}(x, y)\) and a region \(R\) in the \(x, y\) plane are given. Set up and evaluate the double integral that finds the volume between these surfaces over \(R\). \(f_{1}(x, y)=8-x^{2}-y^{2}, f_{2}(x, y)=2 x+y ;\) \(R\) is the square with corners (-1,-1) and (1,1) .
Problem 5
Evaluate the integral and subsequent iterated integral. (a) \(\int_{2}^{3}\left(6 x^{2}+4 x y-3 y^{2}\right) d y\) (b) \(\int_{-3}^{-2} \int_{2}^{5}\left(6 x^{2}+4 x y-3 y^{2}\right) d y d x\)
Problem 6
Points are given in either the rectangular, cylindrical or spherical coordinate systems. Find the coordinates of the points in the other systems. (a) Points in rectangular coordinates: (0,1,1) and (-1,0,1) (b) Points in cylindrical coordinates: \((0, \pi, 1)\) and \((2,4 \pi / 3,0)\) (c) Points in spherical coordinates: \((2, \pi / 6, \pi / 2)\) and \((3, \pi, \pi)\)
Problem 6
Let \(z=f(x, y)\) and \(z=g(x, y)=2 f(x, y) .\) Why is the surface area of \(g\) over a region \(R\) not twice the surface area of \(f\) over \(R ?\)
Problem 6
Evaluate the integral and subsequent iterated integral. (a) \(\int_{0}^{\pi}(2 x \cos y+\sin x) d x\) (b) \(\int_{0}^{\pi / 2} \int_{0}^{\pi}(2 x \cos y+\sin x) d x d y\)
Problem 6
Two surfaces \(f_{1}(x, y)\) and \(f_{2}(x, y)\) and a region \(R\) in the \(x, y\) plane are given. Set up and evaluate the double integral that finds the volume between these surfaces over \(R\). \(f_{1}(x, y)=x^{2}+y^{2}, f_{2}(x, y)=-x^{2}-y^{2}\); \(R\) is the square with corners (0,0) and (2,3) .
Problem 6
(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration. $$ \int_{-\pi / 2}^{\pi / 2} \int_{0}^{\pi}(\sin x \cos y) d x d y $$
Problem 6
Describe a situation where the center of mass of a lamina does not lie within the region of the lamina itself.
Problem 6
In Exercises \(3-10\), a function \(f(x, y)\) is given and a region \(R\) of the \(x-y\) plane is described. Set up and evaluate \(\iint_{R} f(x, y) d A\) using polar coordinates. \(f(x, y)=4 ; R\) is the region enclosed by the petal of the rose curve \(r=\sin (2 \theta)\) in the first quadrant.
Problem 7
In Exercises \(7-10,\) point masses are given along a line or in the plane. Find the center of \(\operatorname{mass} \bar{x}\) or \((\bar{x}, \bar{y}),\) as appropriate. (All masses are in grams and distances are in cm.) $$ m_{1}=4 \text { at } x=1 ; \quad m_{2}=3 \text { at } x=3 ; \quad m_{3}=5 \text { at } x=10 $$
