Problem 14
A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi} \int_{0}^{1} \int_{0}^{2-r} r d z d r d \theta $$
Problem 14
(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 \(R .\) \(\iint_{R} y e^{x} d A,\) where \(R\) is bounded by \(x=0, x=y^{2}\) and \(y=1 .\)
Problem 14
In Exercises \(11-14,\) an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral. $$ \begin{array}{l} \int_{-2}^{-1} \int_{0}^{\sqrt{4-x^{2}}}(x+5) d y d x+\int_{-1}^{1} \int_{\sqrt{1-x^{2}}}^{\sqrt{4-x^{2}}}(x+5) d y d x+ \\ \int_{1}^{2} \int_{0}^{\sqrt{4-x^{2}}}(x+5) d y d x \end{array} $$
Problem 14
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 triangle with corners \((0,0),(1,0),\) and (0,1)\(;\) \(\delta(x, y)=\left(x^{2}+y^{2}+1\right) \mathrm{Ib} / \mathrm{in}^{2}\)
Problem 14
Find the area of the given surface over the region \(R\). \(f(x, y)=-2 x+4 y^{2}+7\) over \(R,\) the triangle bounded by \(y=-x, y=x, 0 \leq y \leq 1\).
Problem 15
A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi} \int_{0}^{3} \int_{0}^{\sqrt{9-r^{2}}} r d z d r d \theta $$
Problem 15
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 disk centered at the origin with radius 2; \(\delta(x, y)=\) \((x+y+4) \mathrm{kg} / \mathrm{m}^{2}\)
Problem 15
Find the area of the given surface over the region \(R\). \(f(x, y)=x^{2}+y\) over \(R,\) the triangle bounded by \(y=2 x\) \(y=0\) and \(x=2\).
Problem 15
In Exercises \(15-16,\) special double integrals are presented that are especially well suited for evaluation in polar coordinates. Consider \(\iint_{f} e^{-\left(x^{2}+y^{2}\right)} d A .\) (a) Why is this integral difficult to evaluate in rectangular coordinates, regardless of the region \(R ?\) (b) Let \(R\) be the region bounded by the circle of radius \(a\) centered at the origin. Evaluate the double integral using polar coordinates. (c) Take the limit of your answer from \((b),\) as \(a \rightarrow \infty\). What does this imply about the volume under the surface of \(e^{-\left(x^{2}+y^{2}\right)}\) over the entire \(x-y\) plane?
Problem 15
(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 \(R .\) \(\iint_{R}(6-3 x-2 y) d A,\) where \(R\) is bounded by \(x=0, y=0\) and \(3 x+2 y=6\)
