Problem 3
Give two uses of triple integration.
Problem 3
Explain why if \(f(x, y)>0\) over a region \(R,\) then \(\iint_{R} f(x, y) d A>0\).
Problem 4
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. $$ \begin{aligned} &f(x, y)=4 x+4 y ; R \text { is the region enclosed by the circle }\\\ &x^{2}+y^{2}=4 \end{aligned} $$
Problem 4
One understanding of an iterated integral is that \(\int_{a}^{b} \int_{g_{1}(x)}^{g_{2}(x)} d y d x\) gives the ______ of a plane region.
Problem 4
If an object has a constant density \(\delta\) and a volume \(V\), what is its mass?
Problem 4
If \(\iint_{R} f(x, y) d A=\iint_{R} g(x, y) d A,\) does this imply \(f(x, y)=\) \(g(x, y) ?\)
Problem 4
Why is it important to know how to set up a double integral to compute surface area, even if the resulting integral is hard to evaluate?
Problem 5
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: (2,2,1) and \((-\sqrt{3}, 1,0)\) (b) Points in cylindrical coordinates: \((2, \pi / 4,2)\) and \((3,3 \pi / 2,-4)\) (c) Points in spherical coordinates: \((2, \pi / 4, \pi / 4)\) and (1,0,0)
Problem 5
(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration. $$ \int_{1}^{2} \int_{-1}^{1}\left(\frac{x}{y}+3\right) d x d y $$
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) .
