Chapter 14

Jacobians and Transformed Regions in the Plane. a. Solve the system $$ u=x-y, \quad v=2 x+y $$ for \(x\) and \(y\) in terms of \(u\) and \(v .\) Then find the value of the Jacobian \(\partial(x, y) / \partial(u, v)\) b. Find the image under the transformation \(u=x-y\) \(v=2 x+y\) of the triangular region with vertices (0,0) \((1,1),\) and (1,-2) in the \(x y\) -plane. Sketch the transformed region in the \(u v\) -plane.

Evaluate the cylindrical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{\sqrt{2-r^{2}}} d z r d r d \theta$$

Find the center of mass of a thin plate of density \(\delta=3\) bounded by the lines \(x=0, y=x,\) and the parabola \(y=2-x^{2}\) in the first quadrant.

Sketch the described regions of integration. $$0 \leq x \leq 3, \quad 0 \leq y \leq 2 x$$

Evaluate the iterated integral. $$\int_{1}^{2} \int_{0}^{4} 2 x y d y d x$$

Jacobians and Transformed Regions in the Plane. a. Solve the system $$ u=x+2 y, \quad v=x-y $$ for \(x\) and \(y\) in terms of \(u\) and \(v .\) Then find the value of the Jacobian \(\partial(x, y) / \partial(u, v)\) b. Find the image under the transformation \(u=x+2 y\) \(v=x-y\) of the triangular region in the \(x y\) -plane bounded by the lines \(y=0, y=x,\) and \(x+2 y=2 .\) Sketch the transformed region in the \(u v\) -plane.

Evaluate the cylindrical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\theta / 2 \pi} \int_{0}^{3+24 r^{2}} d z r d r d \theta$$

Find the moments of inertia about the coordinate axes of a thin rectangular plate of constant density \(\delta\) bounded by the lines \(x=3\) and \(y=3\) in the first quadrant.

Write six different iterated triple integrals for the volume of the rectangular solid in the first octant bounded by the coordinate planes and the planes \(x=1, y=2\) and \(z=3 .\) Evaluate one of the integrals.

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The lines \(x=0, y=2 x,\) and \(y=4\)

Sketch the described regions of integration. $$-1 \leq x \leq 2, \quad x-1 \leq y \leq x^{2}$$

Evaluate the iterated integral. $$\int_{0}^{2} \int_{-1}^{1}(x-y) d y d x$$

Jacobians and Transformed Regions in the Plane. a. Solve the system $$ u=3 x+2 y, \quad v=x+4 y $$ for \(x\) and \(y\) in terms of \(u\) and \(v .\) Then find the value of the Jacobian \(\partial(x, y) / \partial(u, v)\) b. Find the image under the trunsformation \(u=3 x+2 y\) \(v=x+4 y\) of the triangular region in the \(x y\) -plane bounded by the \(x\) -axis, the \(y\) -axis, and the line \(x+y=1\). Sketch the transformed region in the \(u v\) -plane.

Evaluate the cylindrical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\theta / 2 \pi} \int_{0}^{3+24 r^{2}} d z r d r d \theta$$

Find the centroid of the region in the first quadrant bounded by the \(x\) -axis, the parabola \(y^{2}=2 x,\) and the line \(x+y=4\).

Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane \(6 x+3 y+2 z=6 .\) Evaluate one of the integrals.

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The parabola \(x=-y^{2}\) and the line \(y=x+2\)

Sketch the described regions of integration. $$-2 \leq y \leq 2, \quad y^{2} \leq x \leq 4$$

Evaluate the iterated integral. $$\int_{-1}^{0} \int_{-1}^{1}(x+y+1) d x d y$$

a. Solve the system $$ u=2 x-3 y, \quad v=-x+y $$ for \(x\) and \(y\) in terms of \(u\) and \(v .\) Then find the value of the Jacobian \(\partial(x, y) / \partial(u, v)\) b. Find the image under the transformation \(u=2 x-3 y\) \(v=-x+y\) of the parallelogram \(R\) in the \(x y\) -plane with boundaries \(x=-3, x=0, y=x,\) and \(y=x+1 .\) Sketch the transformed region in the \(u v\) -plane.

Evaluate the cylindrical coordinate integrals. $$\int_{0}^{\pi} \int_{0}^{\theta / \pi} \int_{-\sqrt{4-r^{2}}}^{3 \sqrt{4-r^{2}}} z d z r d r d \theta$$

Write six different iterated triple integrals for the volume of the region in the first octant enclosed by the cylinder \(x^{2}+z^{2}=4\) and the plane \(y=3 .\) Evaluate one of the integrals.

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The parabola \(x=y-y^{2}\) and the line \(y=-x\).

Sketch the described regions of integration. $$0 \leq y \leq 1, \quad y \leq x \leq 2 y$$

Evaluate the iterated integral. $$\int_{0}^{1} \int_{0}^{1}\left(1-\frac{x^{2}+y^{2}}{2}\right) d x d y$$

Evaluate the integral $$ \int_{0}^{4} \int_{x=y / 2}^{x=(y / 2)+1} \frac{2 x-y}{2} d x d y $$ from Example 1 directly by integration with respect to \(x\) and \(y\) to confirm that its value is 2.

Evaluate the cylindrical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1 / \sqrt{2-r^{2}}} 3 d z r d r d \theta$$

Let \(D\) be the region bounded by the paraboloids \(z=8-x^{2}-y^{2}\) and \(z=x^{2}+y^{2} .\) Write six different triple iterated integrals for the volume of \(D .\) Evaluate one of the integrals.

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The curve \(y=e^{x}\) and the lines \(y=0, x=0,\) and \(x=\ln 2\).

Sketch the described regions of integration. $$0 \leq x \leq 1, \quad e^{x} \leq y \leq e$$

Evaluate the iterated integral. $$\int_{0}^{3} \int_{0}^{2}\left(4-y^{2}\right) d y d x$$

Evaluate the cylindrical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{1} \int_{-1 / 2}^{1 / 2}\left(r^{2} \sin ^{2} \theta+z^{2}\right) d z r d r d \theta$$

Find the centroid of the region between the \(x\) -axis and the arch \(y=\sin x, 0 \leq x \leq \pi\).

Let \(D\) be the region bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=2 y\) Write triple iterated integrals in the order \(d z d x d y\) and \(d z d y d x\) that give the volume of \(D .\) Do not evaluate either integral.

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The curves \(y=\ln x\) and \(y=2 \ln x\) and the line \(x=e\), in the first quadrant.

Sketch the described regions of integration. $$1 \leq x \leq e^{2}, \quad 0 \leq y \leq \ln x$$

Evaluate the iterated integral. $$\int_{0}^{3} \int_{-2}^{0}\left(x^{2} y-2 x y\right) d y d x$$

The integrals we have seen so far suggest that there are preferred orders of integration for cylindrical coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{0}^{2 \pi} \int_{0}^{3} \int_{0}^{z / 3} r^{3} d r d z d \theta$$

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1}\left(x^{2}+y^{2}+z^{2}\right) d z d y d x$$

Find the moment of inertia about the \(x\) -axis of a thin plate of density \(\delta=1\) bounded by the circle \(x^{2}+y^{2}=4 .\) Then use your result to find \(I_{y}\) and \(I_{0}\) for the plate.

Describe the given region in polar coordinates. The region enclosed by the circle \(x^{2}+y^{2}=2 x\)

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The parabolas \(x=y^{2}\) and \(x=2 y-y^{2}\).

Sketch the described regions of integration. $$0 \leq y \leq 1, \quad 0 \leq x \leq \sin ^{-1} y$$

Evaluate the iterated integral. $$\int_{0}^{1} \int_{0}^{1} \frac{y}{1+x y} d x d y$$

The integrals we have seen so far suggest that there are preferred orders of integration for cylindrical coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{-1}^{1} \int_{0}^{2 \pi} \int_{0}^{1+\cos \theta} 4 r d r d \theta d z$$

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{\sqrt{2}} \int_{0}^{3 y} \int_{x^{2}+3 y^{2}}^{8-x^{2}-y^{2}} d z d x d y$$

Find the moment of inertia with respect to the \(y\) -axis of a thin sheet of constant density \(\delta=1\) bounded by the curve \(y=\left(\sin ^{2} x\right) / x^{2}\) and the interval \(\pi \leq x \leq 2 \pi\) of the \(x\) -axis.

Describe the given region in polar coordinates. The region enclosed by the semicircle \(x^{2}+y^{2}=2 y, y \geq 0\)

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The parabolas \(x=y^{2}-1\) and \(x=2 y^{2}-2\).

Sketch the described regions of integration. $$0 \leq y \leq 8, \quad \frac{1}{4} y \leq x \leq y^{1 / 3}$$