Problem 10
Set up the triple integrals that give the volume of \(D\) in all 6 orders of integration, and find the volume of \(D\) by evaluating the indicated triple integral. \(D\) is bounded by the planes \(y=0, y=2, x=1, z=0\) and \(z=(3-x) / 2\) Evaluate the triple integral with order \(d x d y d z\).
Problem 10
Evaluate the integral and subsequent iterated integral. (a) \(\int_{0}^{x}\left(\frac{1}{1+x^{2}}\right) d y\) (b) \(\int_{1}^{2} \int_{0}^{x}\left(\frac{1}{1+x^{2}}\right) d y d x\)
Problem 10
(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration. $$ \int_{0}^{9} \int_{y / 3}^{\sqrt{y}}\left(x y^{2}\right) d x d y $$
Problem 10
Set up the iterated integral that computes the surface area of the given surface over the region \(R .\) \(f(x, y)=\frac{1}{e^{x^{2}+1}} ; \quad R\) is the rectangle bounded by \(-5 \leq x \leq 5\) and \(0 \leq y \leq 1\).
Problem 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.) $$ \begin{array}{l} m_{1}=1 \text { at }(-1,-1) ; \quad m_{2}=2 \text { at }(-1,1) ; \\ m_{3}=2 \text { at }(1,1) ; \quad m_{4}=1 \text { at }(1,-1) \end{array} $$
Problem 11
A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi / 2} \int_{0}^{2} \int_{0}^{2} r d z d r d \theta $$
Problem 11
In Exercises \(11-18,\) find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the rectangle with corners (1,-3),(1,2),(7,2) and (7,-3)\(; \delta(x, y)=5 \mathrm{gm} / \mathrm{cm}^{2}\)
Problem 11
Find the area of the given surface over the region \(R\). \(f(x, y)=3 x-7 y+2 ; R\) is the rectangle with opposite corners (-1,0) and (1,3).
Problem 11
In Exercises \(11-14,\) an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral. $$ \int_{0}^{5} \int_{-\sqrt{25-x^{2}}}^{\sqrt{25-x^{2}}} \sqrt{x^{2}+y^{2}} d y d x $$
Problem 11
(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R} x^{2} y d A,\) where \(R\) is bounded by \(y=\sqrt{x}\) and \(y=x^{2} .\)
