In 2D integrals the regions of integration (1D domain) are always intervals. When we extend to 3D problems the region (interval) becomes a rectangle (2D domain). However, not all shapes have rectangular bases. The problems below illustrate how to extend the method from Section 3.3 to these non-rectangular domains.
Subsection3.4.1Derivation
Remember that in Activity 53 volume was developed as a sum of areas beneath the curves that constitute the surface. Keep this concept in mind as the integrals over rectangular and non-rectangular domains are compared below.
Activity54.
In this activity we see how bounds need to be modified when they are not rectangular. The first task is a rectangular domain for sake of contrast.
Use \(f(x,y)=2-x^2-y^2\) for the following problems.
(a)
Use the rectangular domain \(x \in [-1,1]\) and \(y \in [-1,1]\) for the following.
(i)
Setup the area integral for the curve from \(f(x,y)\) with \(y=-1.\)
(ii)
What are the left and right boundaries when \(y=-1\text{?}\) Note these are the limits of integration of the area.
(iii)
Setup the area integral for the curve from \(f(x,y)\) with \(y=0.\)
(iv)
What are the left and right boundaries when \(y=0\text{?}\) Note these are the limits of integration of the area.
(v)
Setup the area integral for the curve from \(f(x,y)\) with \(y=1.\)
(vi)
What are the left and right boundaries when \(y=1\text{?}\) Note these are the limits of integration of the area.
(vii)
In general what are the left and right boundaries for any curve from \(f(x,y)\) with a fixed \(y\) value?
(b)
Use the non-rectangular domain \(x^2+y^2=2\) for the following.
(i)
Setup the area integral for the curve from \(f(x,y)\) with \(y=-1.\)
(ii)
What are the left and right boundaries when \(y=-1\text{?}\) Note these are the limits of integration of the area.
(iii)
Setup the area integral for the curve from \(f(x,y)\) with \(y=0.\)
(iv)
What are the left and right boundaries when \(y=0\text{?}\) Note these are the limits of integration of the area.
(v)
Setup the area integral for the curve from \(f(x,y)\) with \(y=1.\)
(vi)
What are the left and right boundaries when \(y=1\text{?}\) Note these are the limits of integration of the area.
(vii)
Setup the area integral for the curve from \(f(x,y)\) with \(y=\sqrt{2}.\)
(viii)
What are the left and right boundaries when \(y=\sqrt{2}\text{?}\) Note these are the limits of integration of the area.
(ix)
In general what are the left and right boundaries for any curve from \(f(x,y)\) with a fixed \(y\) value? Note try repeating the process above with \(y=y_i\) (using \(y\) as a constant).
Figure3.4.1.Integration Over Different Domains
ExercisesExercises
To setup these volume integrals use the process above for finding the \(x\) boundaries given a fixed \(y\) value. Note Fubini’s theorem implies you can swap the roles of \(x\) and \(y\) if it is convenient.
1.
Find the volume between the surface \(s(x,y)=7x+2y+6\) and the \(xy\)-plane over the region \(x \in [-1,1]\) and \(y \in [-1,1].\)
2.
Find the volume between the surface \(s(x,y)=7x+2y+6\) and the \(xy\)-plane over the region between \(x=1-y^2\) and \(x=0.\)
3.
Find the volume between the surface \(s(x,y)=\sin x + \sin y\) and the \(xy\)-plane over the region enclosed by \(y=1,\)\(x=0,\) and \(y=x.\)
4.
Determine the solid whose volume is calculated by
\begin{equation*}
\int_{-2}^1 \int_{x-1}^{1-x^2} x^2+y^2 \; dy \: dx
\end{equation*}
5.
\begin{equation*}
\int_1^4 \int_{\sqrt{y}}^2 \sin\left(\frac{x^3}{3}-x\right) \; dx \: dy
\end{equation*}
Subsection3.4.2Double Integrals with Cylindrical Coordinates
The double integrals developed in Section Subsection 3.3.2 calculate volumes by multiplying the area of a rectangle in the domain by the height of the surface thus calculating the volume of a cuboid (3D rectangle). This same method (area of base times height) can be used for volumes using cylindrical coordinates.
Activity55.
In this activity we see what shape is formed in cylindrical coordinates (as opposed to the cuboid).
Use the surface \(z=r\) for the following problems. The partition for \(r\) is \(\{0,1,2,3\}.\) The partition for \(\theta\) is \(\{0,\pi/2,\pi,3\pi/2\}\text{.}\)
(a)
Construct a table of values for \(z\) using the partitions above.
(b)
Sketch lines from these points orthogonally down to the \(xy\)-plane.
(c)
Review how to calculate an area using polar coordinates.
(d)
Write the integral that will calculate the volume between this surface and the \(xy\)-plane over the region \(r \le 3.\)
To find the volume of one of the shapes complete the following steps.
Activity56.
(a)
Note the object in Figure 3.4.2 consists of two slices of concentric circles.
(b)
Calculate the area of the smaller, circular wedge. If you use an integral, you are working far too hard.
(c)
Calculate the area of the larger, circular wedge.
(d)
Calculate the area between the smaller and larger circular wedges.
(e)
Simplify by dividing, collecting, factoring, and using \(\Delta\theta\) and \(\Delta r\) where appropriate.
(f)
Calculate the limit of this as \(\Delta\theta \to 0 \) and \(\Delta r \to 0.\)
(g)
This is what is integrated in place of the area of a rectangle (i.e., \(dx \: dy\)).
Figure3.4.2.Derivation of Formula for Area in Cylindrical Coordinates
ExercisesExercises
Note Theorem 3.3.3 works regardless of the coordinate system. It states that under the conditions iterated integrals can be used instead of the double integral for two variables without respect to what those variables represent.
1.
Find the volume enclosed between \(z=1+\cos r\) and the \(xy\)-plane for \(r \in [0,2\pi].\)
2.
Find the volume enclosed between \(z=1\) and \(z=r.\)
3.
Find the volume enclosed between \(z=1+\sin\theta\) and the \(xy\)-plane for \(r \in [0,2].\)
