Problem 1
In Problems 1-6, evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. $$ \int_{0}^{2 \pi} \int_{0}^{3} \int_{0}^{12} r d z d r d \theta $$
Problem 1
Evaluate each of the iterated integrals. $$ \int_{0}^{2} \int_{0}^{3}(9-x) d y d x $$
Problem 1
Evaluate the iterated integrals. \(\int_{0}^{\pi / 2} \int_{0}^{\cos \theta} r^{2} \sin \theta d r d \theta\)
Problem 1
Find the mass \(m\) and center of mass \((\bar{x}, \bar{y})\) of the lamina bounded by the given curves and with the indicated density. x=0, x=4, y=0, y=3 ; \delta(x, y)=y+1
Problem 1
For the transformation \(x=u+v, y=v-u\), sketch the \(u\) -curves and \(v\) -curves for the grid \(\\{(u, v):(u=2,3,4,5\) and \(1 \leq v \leq 3)\) or \((v=1,2,3\) and \(2 \leq u \leq 5)\\}\)
Problem 1
Find the area of the indicated surface. Make a sketch in each case. The part of the plane \(3 x+4 y+6 z=12\) that is above the rectangle in the \(x y\) -plane with vertices \((0,0),(2,0),(2,1)\), and \((0,1)\).
Problem 1
Evaluate the iterated integrals in Problems \(1-14 .\) $$ \int_{0}^{1} \int_{0}^{3 x} x^{2} d y d x $$
Problem 1
Evaluate the iterated integrals. \(\int_{-3}^{7} \int_{0}^{2 x} \int_{y}^{x-1} d z d y d x\)
Problem 2
Evaluate the iterated integrals. \(\int_{0}^{\pi / 2} \int_{0}^{\sin \theta} r d r d \theta\)
Problem 2
Find the mass \(m\) and center of mass \((\bar{x}, \bar{y})\) of the lamina bounded by the given curves and with the indicated density. \(y=0, y=\sqrt{4-x^{2}} ; \delta(x, y)=y\)
