Describing regions

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We describe areas and volumes of regions using iterated integrals.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Trivial regions

Without doing any integrals, evaluate $\iint _R \d A$ where $R =\{(x,y):-2\le x\le 4 \text { and } -6\le y\le 1\}.$

Without doing any integrals, evaluate $\iiint _R \d V$ where $R =\{(x,y,z):-4\le x\le 3, -1\le y\le 5,\text { and, } -3\le z\le 2\}.$

Two-dimensional regions

Consider the region $R=\{(x,y):0\le x\le 2+\cos (y)\text { and }0<y<2\pi \}$

Compute the area of this region with an iterated integral where one integrates first with respect to $x$ and next integrates with respect to $y$ .

As a challenge, compute the area with an iterated integral where one integrates first with respect to $y$ and next integrates with respect to $x$ .

Consider the region

(a): Compute the area of this region with an iterated integral where one integrates first with respect to $y$ and next integrates with respect to $x$ .
(b): Compute the area of this region with an iterated integral where one integrates first with respect to $x$ and next integrates with respect to $y$ .

Consider the ellipse centered at the origin:

Recall that the implicit formula for an ellipse is given by $\left (\frac {x-x_0}{a}\right )^2 + \left (\frac {y-y_0}{b}\right )^2 = 1.$

(a): Compute the area of this region with an iterated integral where one integrates first with respect to $y$ and next integrates with respect to $x$ .
(b): Compute the area of this region with an iterated integral where one integrates first with respect to $x$ and next integrates with respect to $y$ .

Consider the following region:

The largest $y$ -value of the region is $y=\sqrt {2}$ .

(a): Compute the area of this region with an iterated integral where one integrates first with respect to $y$ and next integrates with respect to $x$ .
(b): Compute the area of this region with an iterated integral where one integrates first with respect to $x$ and next integrates with respect to $y$ .

Three-dimensional regions

Consider the region bounded by the planes:

$x=0$
$y=0$
$z=0$
$6x+3y+2z = 24$

Sketch this region and set-up six iterated integrals that compute the volume of this solid, all which integrate in a different order.

Consider the region: $R=\{(x,y,z): 0 \le x \le 1, -1\le y \le 0, 0\le z\le y^2\}$

Set up six iterated integrals that compute the volume of this solid, all which integrate in a different order.

Consider the region: $R=\{(x,y,z): 0 \le x \le 1, \sqrt {x}\le y \le 1, 0\le z\le 1-y\}$

Set up six iterated integrals that compute the volume of this solid, all which integrate in a different order.