Problem 12
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)\(; \delta(x, y)=\left(x+y^{2}\right) \mathrm{gm} / \mathrm{cm}^{2}\)
Problem 12
Find the area of the given surface over the region \(R\). \(f(x, y)=2 x+2 y+2 ; R\) is the triangle with corners (0,0) (1,0) and (0,1)
Problem 12
In Exercises \(11-14,\) an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral. $$ \int_{-4}^{4} \int_{-\sqrt{16-y^{2}}}^{0}(2 y-x) d x d y $$
Problem 12
A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{3}^{4} \int_{0}^{5} r d z d r d \theta $$
Problem 12
(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 \(y=\sqrt[3]{x}\) and \(y=x^{3} .\)
Problem 13
In Exercises \(11-14,\) an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral. $$ \int_{0}^{2} \int_{y}^{\sqrt{8-y^{2}}}(x+y) d x d y $$
Problem 13
(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 \(R .\) \(\iint_{R} x^{2}-y^{2} d A,\) where \(R\) is the rectangle with corners (-1,-1),(1,-1),(1,1) and (-1,1)
Problem 13
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 \((-1,0),(1,0),\) and (0,1)\(;\) \(\delta(x, y)=2 \mathrm{lb} / \mathrm{in}^{2}\)
Problem 13
A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{1} \int_{0}^{1-r} r d z d r d \theta $$
Problem 13
Find the area of the given surface over the region \(R\). \(f(x, y)=x^{2}+y^{2}+10 ; R\) is bounded by the circle \(x^{2}+y^{2}=\) \(16 .\)
