91Ó°ÊÓ

Find the limits on \(\iint d y d x\) and \(\iint d x d y\). Draw \(R\) and compute its area. $$ R=\text { triangle inside the lines } y=x, y=-x, y=3 $$

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). $$ x=e^{2 u+v}, y=e^{u+2 v} $$

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). $$ x=u v, y=v^{2}-u^{2} $$

Find the limits on \(\iint d y d x\) and \(\iint d x d y\). Draw \(R\) and compute its area. $$ R=\text { triangle inside the lines } y=x, y=2 x, y=4 $$

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

Find the limits on \(\iint d y d x\) and \(\iint d x d y\). Draw \(R\) and compute its area. $$ R=\text { triangle with vertices }(0,0),(4,4),(4,8) $$

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). $$ x=u, y=v\left(l+u^{2}\right) $$

Find the limits in \(\iiint d x d y d z\) or \(\iiint d z d y d x\). Compute the volume. A cube is inscribed in a sphere: radius 1, both centers at (0,0,0). What is the volume of the cube?

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

Find the volume and the centroid of the region bounded by x =0, y =0, z = 0, and x/a + y/b + z/c = 1.