Problem 7
(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration. $$ \int_{0}^{4} \int_{0}^{-x / 2+2}\left(3 x^{2}-y+2\right) d y d x $$
Problem 7
Describe the curve, surface or region in space determined by the given bounds. Bounds in cylindrical coordinates: (a) \(r=1, \quad 0 \leq \theta \leq 2 \pi, \quad 0 \leq z \leq 1\) (b) \(1 \leq r \leq 2, \quad 0 \leq \theta \leq \pi, \quad 0 \leq z \leq 1\) Bounds in spherical coordinates: (c) \(\rho=3, \quad 0 \leq \theta \leq 2 \pi, \quad 0 \leq \varphi \leq \pi / 2\) (d) \(2 \leq \rho \leq 3, \quad 0 \leq \theta \leq 2 \pi, \quad 0 \leq \varphi \leq \pi\)
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.) $$ m_{1}=4 \text { at } x=1 ; \quad m_{2}=3 \text { at } x=3 ; \quad m_{3}=5 \text { at } x=10 $$
Problem 7
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 \(R\). \(f_{1}(x, y)=\sin x \cos y, f_{2}(x, y)=\cos x \sin y+2\); \(R\) is the triangle with corners \((0,0),(\pi, 0)\) and \((\pi, \pi)\).
Problem 7
Set up the iterated integral that computes the surface area of the given surface over the region \(R .\) \(f(x, y)=\sin x \cos y ; \quad R\) is the rectangle with bounds \(0 \leq\) \(x \leq 2 \pi, \quad 0 \leq y \leq 2 \pi\).
Problem 7
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. $$ \begin{aligned} &f(x, y)=\ln \left(x^{2}+y^{2}\right) ; R \text { is the annulus enclosed by the cir- }\\\ &\text { cles } x^{2}+y^{2}=1 \text { and } x^{2}+y^{2}=4 \end{aligned} $$
Problem 8
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. $$ \begin{aligned} &f(x, y)=1-x^{2}-y^{2} ; R \text { is the region enclosed by the circle }\\\ &x^{2}+y^{2}=1 \end{aligned} $$
Problem 8
Set up the iterated integral that computes the surface area of the given surface over the region \(R .\) \(f(x, y)=\frac{1}{x^{2}+y^{2}+1} ; \quad R\) is bounded by the circle \(x^{2}+\) \(y^{2}=9\).
Problem 8
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.) $$ \begin{array}{l} m_{1}=2 \text { at } x=-3 ; \quad m_{2}=2 \text { at } x=-1; \\ m_{3}=3 \text { at } x=0 ; \quad m_{4}=3 \text { at } x=7 \end{array} $$
Problem 8
Describe the curve, surface or region in space determined by the given bounds. Bounds in cylindrical coordinates: (a) \(1 \leq r \leq 2, \quad \theta=\pi / 2, \quad 0 \leq z \leq 1\) (b) \(r=2, \quad 0 \leq \theta \leq 2 \pi, \quad z=5\) Bounds in spherical coordinates: (c) \(0 \leq \rho \leq 2, \quad 0 \leq \theta \leq \pi, \quad \varphi=\pi / 4\) (d) \(\rho=2, \quad 0 \leq \theta \leq 2 \pi, \quad \varphi=\pi / 6\)
