Problem 16
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 coordinate planes and by \(z=1-y / 3\) and \(z=1-x\) Evaluate the triple integral with order \(d x d y d z\).
Problem 16
A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-r^{2}}+b} r d z d r d \theta $$
Problem 16
Find the area of the given surface over the region \(R\). \(f(x, y)=\frac{2}{3} x^{3 / 2}+2 y^{3 / 2}\) over \(R,\) the rectangle with opposite corners (0,0) and (1,1).
Problem 16
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 circle sector bounded by \(x^{2}+y^{2}=25\) in the first quadrant; \(\delta(x, y)=\left(\sqrt{x^{2}+y^{2}}+1\right) \mathrm{kg} / \mathrm{m}^{2}\)
Problem 16
(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 \(R .\) \(\iint_{R} e^{y} d A,\) where \(R\) is bounded by \(y=\ln x\) and \(y=\frac{1}{e-1}(x-1)\).
Problem 16
In Exercises \(15-16,\) special double integrals are presented that are especially well suited for evaluation in polar coordinates. The surface of a right circular cone with height \(h\) and base radius \(a\) can be described by the equation \(f(x, y)=\) \(h-h \sqrt{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}},\) where the tip of the cone lies at \((0,0, h)\) and the circular base lies in the \(x\) -y plane, centered at the origin. Confirm that the volume of a right circular cone with height \(h\) and base radius \(a\) is \(V=\frac{1}{3} \pi a^{2} h\) by evaluating \(\iint_{R} f(x, y) d A\) in polar coordinates.
Problem 17
(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 \(R .\) \(\iint_{R}\left(x^{3} y-x\right) d A,\) where \(R\) is the half of the circle \(x^{2}+y^{2}=9\) in the first and second quadrants.
Problem 17
A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{1} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$
Problem 17
Iterated integrals are given that compute the area of a region \(R\) in the \(x\) -y plane. Sketch the region \(R\), and give the iterated integral(s) that give the area of \(R\) with the opposite order of integration. $$ \int_{-2}^{2} \int_{0}^{4-x^{2}} d y d x $$
Problem 17
Find the area of the given surface over the region \(R\). \(f(x, y)=10-2 \sqrt{x^{2}+y^{2}}\) over \(R,\) bounded by the circle \(x^{2}+y^{2}=25\). (This is the cone with height 10 and base radius 5 ; be sure to compare your result with the known formula.)
