91Ó°ÊÓ

From the limits of integration deserihe each region and find its volume. The inner integral has the inner limits. $$ \int_{e=0}^{2 \pi} \int_{z=0}^{1} \int_{r=0}^{2-z} r d r d z d \theta $$

Find the limits of integration in \(\iiint d x d y d z\) and the volume of solids. Draw a very rough picture. The volume above \(z=0\) below the cone \(\sqrt{x^{2}+y^{2}}=1-z\).

From the limits of integration deserihe each region and find its volume. The inner integral has the inner limits. $$ \int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{1} \rho^{2} \sin \phi d \rho d \phi d \theta $$

From the limits of integration deserihe each region and find its volume. The inner integral has the inner limits. $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 3} \int_{\sec \phi}^{2} \rho^{2} \sin \phi d \rho d \phi d \theta $$

Find the limits in \(\iiint d x d y d z\) or \(\iiint d z d y d x\). Compute the volume. A circular cylinder with height 6 and base \(x^{2}+y^{2} \leqslant 1\).

From the limits of integration deserihe each region and find its volume. The inner integral has the inner limits. $$ \int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\sin \phi} \rho^{2} \sin \phi d \rho d \phi d \theta $$

Draw the \(x y\) region \(R\) that corresponds to the \(u v\) square \(S\) with corners (0,0),(1,0),(0,1),(1,1) . Locate the corners of \(R\) and then its sides (bike a jigsaw puzzle). $$ x=2 u+v, y=u+2 v $$

Find the limits on \(\iint d y d x\) and \(\iint d x d y\). Draw \(R\) and compute its area. $$ R=\text { triangle inside the lines } x=0, y=1, y=2 x $$

From the limits of integration deserihe each region and find its volume. The inner integral has the inner limits. $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{1}^{3} \rho^{2} \sin \phi d \rho d \phi d \theta $$

Draw the \(x y\) region \(R\) that corresponds to the \(u v\) square \(S\) with corners (0,0),(1,0),(0,1),(1,1) . Locate the corners of \(R\) and then its sides (bike a jigsaw puzzle). $$ x=3 u+2 v, y=u+v $$