Chapter 13

Suppose \(S\) is the unit square in the first quadrant of the \(u v\) -plane. Describe the image of the transformation \(T: x=2 u, y=2 v\)

Explain how cylindrical coordinates are used to describe a point in \(\mathbb{R}^{3}\).

Explain how to find the balance point for two people on opposite ends of a (massless) plank that rests on a pivot.

$$\text { Sketch the region } D=\left\\{(x, y, z): x^{2}+y^{2} \leq 4,0 \leq z \leq 4\right\\}$$

Write an iterated integral that gives the volume of the solid bounded by the surface \(f(x, y)=x y\) over the square \(R=\\{(x, y): 0 \leq x \leq 2,1 \leq y \leq 3\\}\)

Draw the region \(\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq \pi / 2\\} .\) Why is it called a polar rectangle?

Explain how to compute the Jacobian of the transformation \(T: x=g(u, v), y=h(u, v)\)

Explain how spherical coordinates are used to describe a point in \(\mathbb{R}^{3}\).

If a thin \(1-\mathrm{m}\) cylindrical rod has a density of \(\rho=1 \mathrm{g} / \mathrm{cm}\) for its left half and a density of \(\rho=2 \mathrm{g} / \mathrm{cm}\) for its right half, what is its mass and where is its center of mass?

Write an iterated integral for \(\iiint_{D} f(x, y, z) d V,\) where \(D\) is the box \(\\{(x, y, z): 0 \leq x \leq 3,0 \leq y \leq 6,0 \leq z \leq 4\\}\)

Write an iterated integral that gives the volume of a box with height 10 and base \(R=\\{(x, y): 0 \leq x \leq 5,-2 \leq y \leq 4\\}\)

Write the double integral \(\iint_{R} f(x, y) d A\) as an iterated integral in polar coordinates when \(R=\\{(r, \theta): a \leq r \leq b, \alpha \leq \theta \leq \beta\\}\)

Describe and a sketch a region that is bounded on the left and on the right by two curves.

Using the transformation \(T: x=u+v, y=u-v,\) the image of the unit square \(S=\\{(u, v): 0 \leq u \leq 1,0 \leq v \leq 1\\}\) is a region \(R\) in the \(x y\) -plane. Explain how to change variables in the integral \(\iint_{R} f(x, y) d A\) to find a new integral over \(S\).

Explain how to find the center of mass of a thin plate with a variable density.

Write two iterated integrals that equal \(\iint_{R} f(x, y) d A,\) where \(R=\\{(x, y):-2 \leq x \leq 4,1 \leq y \leq 5\\}\)

Sketch the region of integration for the integral \(\int_{-\pi / 6}^{\pi / 6} \int_{1 / 2}^{\cos 2 \theta} f(r, \theta) r d r d \theta\)

Which order of integration is preferable to integrate \(f(x, y)=x y\) over \(R=\\{(x, y): y-1 \leq x \leq 1-y, 0 \leq y \leq 1\\} ?\)

Suppose \(S\) is the unit cube in the first octant of \(u v w\) -space with one vertex at the origin. What is the image of the transformation \(T: x=u / 2, y=v / 2, z=w / 2 ?\)

In the integral for the moment \(M_{x}\) of a thin plate, why does \(y\) appear in the integrand?

Describe the set \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) in spherical coordinates.

Sketch the region of integration for the integral \(\int_{0}^{1} \int_{0}^{\sqrt{1-z^{2}}} \int_{0}^{\sqrt{1-y^{2}-z^{2}}} f(x, y, z) d x d y d z\)

Consider the integral \(\int_{1}^{3} \int_{-1}^{1}\left(2 y^{2}+x y\right) d y d x .\) State the variable of integration in the first (inner) integral and the limits of integration. State the variable of integration in the second (outer) integral and the limits of integration.

Explain why the element of area in Cartesian coordinates \(d x d y\) becomes \(r d r d \theta\) in polar coordinates.

Which order of integration would you use to find the area of the region bounded by the \(x\) -axis and the lines \(y=2 x+3\) and \(y=3 x-4\) using a double integral?

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=2 u, y=v / 2$$

Explain how to find the center of mass of a three-dimensional object with a variable density.

Explain why \(d z r d r d \theta\) is the volume of a small "box" in cylindrical coordinates.

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

How do you find the area of a region \(R=\\{(r, \theta): 0 \leq g(\theta) \leq r \leq h(\theta), \alpha \leq \theta \leq \beta\\} ?\)

Change the order of integration in the integral \(\int_{0}^{1} \int_{y^{2}}^{\sqrt{y}} f(x, y) d x d y\).

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=-u, y=-v$$

In the integral for the moment \(M_{x z}\) with respect to the \(x z\) -plane of a solid, why does \(y\) appear in the integrand?

Explain why \(\rho^{2} \sin \varphi d \rho d \varphi d \theta\) is the volume of a small "box" in spherical coordinates.

Write an integral for the average value of \(f(x, y, z)=x y z\) over the region bounded by the paraboloid \(z=9-x^{2}-y^{2}\) and the xy-plane (assuming the volume of the region is known).

Evaluate the following iterated integrals. $$\int_{1}^{2} \int_{0}^{1}\left(3 x^{2}+4 y^{3}\right) d y d x$$

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=(u+v) / 2, y=(u-v) / 2$$

Sketch the following systems on a number line and find the location of the center of mass. $$m_{1}=10 \mathrm{kg} \text { located at } x=3 \mathrm{m} ; m_{2}=3 \mathrm{kg} \text { located at } x=-1 \mathrm{m}$$

Write the integral \(\iiint_{D} f(r, \theta, z) d V\) as an iterated integral where \(D=\\{(r, \theta, z): G(r, \theta) \leq z \leq H(r, \theta), g(\theta) \leq r \leq h(\theta)\) \(\alpha \leq \theta \leq \beta\\}\).

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\int_{-2}^{2} \int_{3}^{6} \int_{0}^{2} d x d y d z$$

Evaluate the following iterated integrals. $$\int_{1}^{3} \int_{0}^{2} x^{2} y d x d y$$

Sketch the following polar rectangles. $$R=\\{(r, \theta): 0 \leq r \leq 5,0 \leq \theta \leq \pi / 2\\}$$

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=2 u+v, y=2 u$$

Sketch the following systems on a number line and find the location of the center of mass. $$\begin{aligned} &m_{1}=8 \mathrm{kg} \text { located at } x=2 \mathrm{m} ; m_{2}=4 \mathrm{kg} \text { located at } x=-4 \mathrm{m}\\\ &m_{3}=1 \mathrm{kg} \text { located at } x=0 \mathrm{m} \end{aligned}$$

Write the integral \(\iiint_{D} f(\rho, \varphi, \theta) d V\) as an iterated integral, where \(D=\\{(\rho, \varphi, \theta): g(\varphi, \theta) \leq \rho \leq h(\varphi, \theta), a \leq \varphi \leq b\) \(\alpha \leq \theta \leq \beta\\}\)

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\int_{-1}^{1} \int_{-1}^{2} \int_{0}^{1} 6 x y z d y d x d z$$

Evaluate the following iterated integrals. $$\int_{0}^{3} \int_{-2}^{1}(2 x+3 y) d x d y$$

Sketch the following polar rectangles. $$R=\\{(r, \theta): 2 \leq r \leq 3, \pi / 4 \leq \theta \leq 5 \pi / 4\\}$$

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=u^{2}-v^{2}, y=2 u v$$

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\int_{-2}^{2} \int_{1}^{2} \int_{1}^{e} \frac{x y^{2}}{z} d z d x d y$$