Week 8: The Riemann Integral in \(n\)-D

Key Terms

• The Riemann integral for scalar functions of \(n\) variables

• The Riemann integral for vector functions

• Change-of-variables theorem (also known as the transformation theorem): Coordinate change in \(\mathbb{R}^n\)

• The Jacobian determinant

• Typical coordinates…

  • in \(\mathbb{R}^2\): Cartesian and polar coordinates

  • in \(\mathbb{R}^3\): Cartesian, spherical, cylindrical/semi-polar coordinates

Preparation and Syllabus

• Reading material: The rest of Chapter 6

• Python demo

---

Exercises – Long Day

1: Plane Integrals over Rectangles. By Hand

Question a

Consider the region \(B=\left\lbrace (x,y) \bigm| 0\leq x\leq 2 \wedge -1\leq y\leq 0\right\rbrace\) in \(\mathbb{R}^2\). Compute the plane integral

\[\begin{equation*} \int_B (x^2y^2+x) \mathrm{d}\pmb{x} \end{equation*}\]

using the formula for double integrals over (axis-parallel) rectangles.

Question b

We want to compute the same plane integral once more, but now in what at first sight appears to be a more complicated manner, which is to use the change-of-variables theorem for integrals over \(\mathbb{R}^2\).

Hint

You must first determine a parametrization of the region. Notice that \(B\) is an axis-parallel rectangle in the the \((x,y)\) plane.

Hint

A possible parametrization is \(\pmb{r}(u,v)=(2u,v-1)\) where \((u,v) \in [0,1]^2\). An alternative is \(\pmb{r}(u,v)=(2u,-v)\) where \((u,v) \in [0,1]^2\). Maybe try them both.

Hint

Find (the absolute value of) the Jacobian determinant. Then express the integrand \(x^2y^2+x\) by \(u\) and \(v\).

Hint

Remember the relation \((x,y) = \pmb{r}(u,v)\).

Answer

\[\begin{equation*} \int_B x^2y^2+x \;\mathrm{d}\pmb{x}=\frac{26}{9} \end{equation*}\]

Question c

Compute the plane integral

\[\begin{equation*} \int_B \frac{y}{1+xy} \;\mathrm{d}\pmb{x}, \quad\text{where}\quad B=\left\lbrace (x,y) \mid 0\leq x\leq 1 \, \wedge \, 0\leq y\leq 1\right\rbrace. \end{equation*}\]

Hint

Note that \(B\) again is an axis-parallel rectangle in the \((x,y)\) plane.

Hint

\[\begin{equation*} \int_B \left(\frac{y}{1+xy}\right) \;\mathrm{d}\pmb{x}=\int_{0}^{1} \left(\int_{0}^{1} \frac{v}{1+uv} \mathrm{d}u \right)\mathrm{d}v. \end{equation*}\]

Hint

For the inner integral, use the substitution method by choosing the inner function as

\[\begin{equation*} t=g(u)=1+uv \end{equation*}\]

and the outer one as

\[\begin{equation*} f(t)=\displaystyle{\frac 1t}. \end{equation*}\]

Answer

\[\begin{equation*} \int_B \frac{y}{1+xy} \;\mathrm{d}\pmb{x}=2\ln 2-1 \end{equation*}\]

2: Polar Coordinates. By Hand

A function \(f:\mathbb{R}^2 \to \mathbb{R}\) is given by

\[\begin{equation*} f(x,y)=x^2-y^2. \end{equation*}\]

For a given point \(\pmb{x}=(x,y)\) in the plane, \(r = \Vert \pmb{x} \Vert\) denotes the distance from the point to the origin \((0,0)\). Similarly, \(\theta\) denotes the angle between the \(x\) axis and the position vector to the point, considered with a sign with counterclockwise being the positive direction. A set of points \(B\) is in polar coordinates described as the points for which

\[\begin{equation*} 0\leq r \leq a \, \text{ and } \, -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{2}, \end{equation*}\]

where \(a\) is an arbitrary positive, real number.

Question a

Draw a sketch of \(B\), and compute the area of \(B\) both using integration as well as via elementary geometric considerations.

Hint

You can use this parametric representation of \(B\): \((x,y)=\pmb{p}(u,v)=(u\cos(v),u\sin (v))\) for \(-\frac{\pi}{4} \leq v \leq \frac{\pi}{2}\) and \(0\leq u \leq a\). Here you may of course rename \(u\) to \(r\), and \(v\) to \(\theta\) (which is what is typically used when dealing with polar coordinates).

Answer

The area is \(\frac{3}{8} a^2\pi\).

Question b

Compute the plane integral \(\int_B f(x,y) \;\mathrm{d}\pmb{x}\).

Hint

By hand, you may find the following trigonometric formula for doubled angles useful when finding anti-derivatives:

\[\begin{equation*} \cos^2(v)-\sin^2(v)=\cos(2v). \end{equation*}\]

Answer

\[\begin{equation*} \int_B (x^2-y^2) \;\mathrm{d}\pmb{x}=\frac{a^4}{8} \end{equation*}\]

3: Volume of a Parallelotope

A parallelotope \(P\) in \(\mathbb{R}^n\) “spanned” by the vectors \(\pmb{a}_1, \pmb{a}_2, \dots, \pmb{a}_n\) is defined by:

\[\begin{equation*} P = \left\{ \pmb{y} \in \mathbb{R}^n \mid \, \pmb{y} = A\pmb{x}, \quad \text{where } x_i \in [0,1] \text{ for $i=1,2,\dots, n$} \right\}, \end{equation*}\]

where \(A = [\pmb{a}_1 | \pmb{a}_2 | \cdots | \pmb{a}_n]\) is the \(n \times n\) matrix whose \(i\)’th column is \(\pmb{a}_i\). This set of points can in short be written as \(P=A([0,1]^n)\).

It can be shown via tools solely from Mathematics 1a (in particular the characterization of the determinant) that the \(n\)-dimensional volume of \(P\) is:

\[\begin{equation*} \mathrm{vol}_n(P) = |\mathrm{det}(A)|. \end{equation*}\]

(For the interested student, a proof of this is found here: https://textbooks.math.gatech.edu/ila/determinants-volumes.html .)

In \(\mathbb{R}^2\), a parallelotope is the well-known parallelogram, and \(\mathrm{vol}_n(P)\) is the area of \(P\), while we in \(\mathbb{R}^3\) find the parallelepiped with the usual volume.

Question a

Show \(\mathrm{vol}_n(P) = |\mathrm{det}(A)|\) by use of the change-of-variables theorem for integrals over \(\mathbb{R}^n\).

Hint

Since \(P=A([0,1]^n)\), we will choose \(\Gamma = [0,1]^n\). What is the corresponding parametrization?

Hint

\(\pmb{r}(\pmb{u}) = A \pmb{u}\) where \(\pmb{u} \in \Gamma = [0,1]^n\). What is the corresponding Jacobian matrix?

Answer

The Jacobian determinant is \(\mathrm{det}(A)\). For computation of \(\mathrm{vol}_n(P)\) one must use \(f(\pmb{x}) = 1\). The full integrand is hence the constant \(|\mathrm{det}(A)|\) which is to be integrated over \(\Gamma = [0,1]^n\), which simply becomes \(|\mathrm{det}(A)|\). The integral we have computed is by definition equal to \(\mathrm{vol}_n(P)\).

In the rest of this exercise we wish to investigate the proposition \(\mathrm{vol}_n(P) = |\mathrm{det}(A)|\) without use of integration techniques.

Question b

Let \(n=2\). Choose two linearly independent vectors \(\pmb{a}_1, \pmb{a}_2\) in \(\mathbb{R}^2\). For example, you could choose \(\pmb{a}_1 \in \mathrm{span}(\pmb{e}_1)\). Compute (via simple geometric considerations) the area of the parallelogram “spanned” by the two vectors. Also compute \(|\mathrm{det}(A)|\) and compare the sizes.

Hint

You might need to do the orthogonal projection of \(\pmb{a}_2\) onto the orthogonal complement to \(\pmb{a}_1\).

Question c

Let \(n=2\) and now let \(\pmb{a}_1, \pmb{a}_2\) be arbitrary but linearly independent vectors in \(\mathbb{R}^2\). Can you prove the formula \(\mathrm{area}(P) = |\mathrm{det}(A)|\), where \(P\) is the parallelogram “spanned” by the two vectors? You may assume (why?) that \(\pmb{a}_1 \in \mathrm{span}(\pmb{e}_1)\) if that helps your argument.

Hint

You might again need to make the orthogonal projection of \(\pmb{a}_2\) onto the orthogonal complement to \(\pmb{a}_1\).

Answer

We here provide a proof without use of trigonometric identities:

Since rotation does not change the area of a region, we can by rotating the parallelogram assume that \(\pmb{a}_1 \in \mathrm{span}(\pmb{e}_1)\), meaning that \(A\) is an upper triangular matrix \(A = [a_{i,j}]\) with \(a_{1,2}=0\). Since \(A\) is an upper triangular matrix, the determinant is the product of the diagonal elements, so \(\mathrm{det}(A) = a_{1,1} a_{2,2}\). Hence we have \(|\mathrm{det}(A)| = |a_{1,1} a_{2,2}|\). We now just need to show that this also is the area.

The area of the parallelogram is given by the length of \(\pmb{a}_1\) (meaning \(|a_{1,1}|\)) multiplied by the “height”, which is the length of the orthogonal projection of \(\pmb{a}_2\) onto \(\mathrm{span}(\pmb{e}_2)\) (so \(|\langle \pmb{a}_2, \pmb{e}_2 \rangle| = |a_{2,2}|\)). Hence, the area is \(|a_{1,1}| |a_{2,2}| = |a_{1,1} a_{2,2}|\).

Question d

Let \(n=3\). Choose three linearly independent vectors \(\pmb{a}_1, \pmb{a}_2, \pmb{a}_3\) in \(\mathbb{R}^3\). It might be beneficial to choose \(\pmb{a}_1, \pmb{a}_2 \in \mathrm{span}(\pmb{e}_1, \pmb{e}_2)\). Compute (using simple geometric considerations) the volume of the parallelepiped that is “spanned” by the three vectors. Also compute \(|\mathrm{det}(A)|\) and compare the sizes.

Hint

You might need to make an orthogonal projection of \(\pmb{a}_2\) onto the orthogonal complement to \(\pmb{a}_1\).

Question e (optional/extra)

Let \(n=3\), and now let \(\pmb{a}_1, \pmb{a}_2, \pmb{a}_3\) be arbitrary but linearly independent vectors in \(\mathbb{R}^3\). Can you prove the formula \(\mathrm{area}(P) = |\mathrm{det}(A)|\), where \(P\) is the parallelepiped that is “spanned” by the three vectors? You may assume (why?) that \(\pmb{a}_1, \pmb{a}_2 \in \mathrm{span}(\pmb{e}_1, \pmb{e}_2)\), if that makes your argument easier.

4: Plane Integral over Parametrized Region I

In the \((x,y)\) plane we are given the point \(P_0=(1,2)\) as well as the set of points

\[\begin{equation*} C=\left\lbrace (x,y)\Big\vert \frac 32\leq y \leq \frac 52 \wedge 0\leq x\leq \frac 12 y^2\right\rbrace. \end{equation*}\]

Question a

Draw a temporary sketch of \(C\) and provide a parametric representation \(\pmb{r}(u,v)\) of \(C\) with fitting intervals for \(u\) and \(v\), meaning choose \(\Gamma\) such that \(\pmb{r}(\Gamma)=C\). Argue that the chosen parametrization is injective (if the chosen parametrization is not injective, then find another one).

Question b

Determine the two parameter values \(u_0\) and \(v_0\) such that \(\pmb{r}(u_0,v_0)=P_0\). Make an illustration of \(C\) (e.g. using SymPy) where you place the tangent vectors \(\pmb{r}'_u(u_0,v_0)\) and \(\pmb{r}'_v(u_0,v_0)\) with their starting points at \(P_0\). Determine the area of the parallelogram that is spanned by the tangent vectors (see Exercise 3: Volume of a Parallelotope).

Question c

Determine the Jacobian determinant that corresponds to \(\pmb{r}(u,v)\) and argue that the two column vectors in the Jacobian matrix are linearly independent for all \((u,v) \in \Gamma\). Compute the Jacobian determinant at the point \((u_0,v_0)\).

Question d

Compute the plane integral:

\[\begin{equation*} \int_C \frac{1}{y^2+x} \mathrm{d}\pmb{x} \end{equation*}\]

via the change-of-variables theorem for integrals over \(\mathbb{R}^2\). You must argue for why the change-of-variables theorem can be used.

Answer

A parametrization can for instance be \(\pmb{r}(u,v)=(\tfrac{1}{2}vu^2,u)\) where \(u\in\left[ \tfrac 32,\tfrac 52\right]\) and \(v\in\left[ 0,1\right]\). The corresponding Jacobian determinant is then \(-\frac{1}{2}u^2\).

\[\begin{equation*} \int_C \frac{1}{x^2+y} \mathrm{d}\pmb{x}=\ln(3)-\ln(2). \end{equation*}\]

5: Plane Integral over Parametrized Region II

We want to compute the plane integral

\[\begin{equation*} \int_B 2xy\,\mathrm{d} \pmb{x} \quad\text{where}\quad B=\left\lbrace (x,y) \mid 0\leq x \, \wedge \, 0\leq y, x+y\leq 1\right\rbrace. \end{equation*}\]

Follow the steps below.

Question a

First, sketch the region \(B\). Then determine a parametric representation of \(B\).

Hint

See the parametrization example in the Python demo.

Answer

\(B\) can be parametrized in (infinitely) many ways. One way is:

\[\begin{equation*} \pmb r(u,v)=(u,v(1-u)),\;\text{where}\; u\in\left[ 0,1\right]\text{ and } v\in\left[ 0,1\right]. \end{equation*}\]

Question b

Determine the Jacobian determinant that corresponds to this parametrization. Is the Jacobian determinant different from zero in the interior of the parameter region (which is a requirement from the change-of-variables theorem)?

Answer

\(\mathrm{det}(\pmb{J}_{\pmb{r}}(u,v))=1-u\).

Question c

Now compute the wanted integral.

Hint

You must insert the first and second coordinates of the parametrization into the function \(2xy,\) multiply by (the absolute value of) the Jacobian determinant, and then integrate, first with respect to \(u\) and then to \(v\).

Hint

You must integrate

\[\begin{equation*} \int_0^1\left(\int_0^1 2u^3v-4u^2v+2uv\;\mathrm{d}u\right)\mathrm{d}v. \end{equation*}\]

Hint

An anti-derivative with respect to \(u\) is:

\[\begin{equation*} {\frac 12u^4v-\frac 43u^3v+u^2v}. \end{equation*}\]

When you insert the \(u\) limits, you will get \(\displaystyle{\frac 16 v}\), which you now must integrate with respect to \(v\).

Answer

\[\begin{equation*} \int_B 2xy \,\mathrm{d} \pmb{x}=\frac{1}{12} \end{equation*}\]

6: Integration by Parts and by Substitution in Two Variables

Question a

Determine \(\displaystyle{\int_0^{\frac{\pi}{2}}\left(\int_0^{\frac{\pi}{2}} u\cos(u+v)\mathrm{d}u\right)\mathrm{d}v.}\)

Hint

Determine, using integration by parts with respect to \(u\), an anti-derivative \(F(u)\) for the function

\[\begin{equation*} u\cos(u+v). \end{equation*}\]

Then

\[\begin{equation*} {F(\frac{\pi}{2})-F(0)} \end{equation*}\]

will be a function of \(v\) that we now must find an anti-derivative \(G(v)\) for. Finally, the \(v\) limits are to be inserted.

Hint

\(G(v)=\frac{\pi}{2}\cos(v)-\sin(v)-\cos(v)\)

Answer

\(\displaystyle{\frac{\pi}{2}-2}\)

Question b

Compute \(\displaystyle{\int_0^1\left(\int_0^1 \frac{v}{(uv+1)^2}\mathrm{d}u\right)\mathrm{d}v.}\)

Hint

Find, using integration by substitution with respect to \(u\), an anti-derivative \(F(u)\) for the function

\[\begin{equation*} {\frac{v}{(uv+1)^2}.} \end{equation*}\]

Then

\[\begin{equation*} {F(1)-F(0)} \end{equation*}\]

will be a function of \(v\) that you now must find an anti-derivative \(G(v)\) for. Finalize it by inserting the \(v\) limits.

Hint

\(\displaystyle{G(v)=1-\frac{1}{v+1}}\)

Answer

\(1-\ln(2)\)

7: A Triple Integral

Compute the triple integral

\[\begin{equation*} \displaystyle{\int_1^2\int_1^2\int_1^2 \frac{xy}{z} \,\mathrm dx\,\mathrm dy\,\mathrm dz.}\ \end{equation*}\]

Hint

We are integrating over an axis-parallel box!

Answer

\(\displaystyle{\frac 94 \ln(2)}\)

---

Exercises – Short Day

1: Parametrized Spatial Region. By Hand.

A region \(B\) in \((x,y,z)\) space is given by the parametric representation

\[\begin{equation*} \pmb{r}(u,v,w)=\big(\frac{1}{2}u^2-v^2,-uv,w\big),\quad u\in \left[ 0,2\right],v\in \left[ 0,2\right],w\in \left[ 0,2\right]. \end{equation*}\]

Question a

Within \(B\) we are given the point

\[\begin{equation*} \pmb{x}_0=\pmb{r}(1,1,1). \end{equation*}\]

Find \(\pmb{x}_0\). From the initial point \(\pmb{x}_0\), the tangent vectors \(\pmb{r}_u'(1,1,1),\pmb{r}_v'(1,1,1),\) and \(\pmb{r}_w'(1,1,1)\) span a parallelepiped \(P\), as was described in Long-Day Exercise 3: Volume of a Parallelotope. Compute the volume of this parallelepiped. Illustrate with SymPy.

Hint

The volume is found as the absolute value of the determinant of the matrix that has the spanning vectors as columns.

Answer

\(\pmb{x}_0=(-1/2, -1, 1)\). The volume of \(P\) is \(3\).

Question b

Compute the absolute value of the Jacobian determinant that corresponds to \(\pmb{r}\). Evaluate it at \(\pmb{x}_0\).

Hint

The Jacobian determinant is the determinant of the Jacobian matrix. Why is this matrix related to what we found in the previous question?

Answer

\(|\pmb{J}_{\pmb{r}}(u,v,w)|=u^2+2v^2\). So, we have \(|\pmb{J}_{\pmb{r}}(1,1,1)|=1^2+2\cdot 1^2 = 3\).

Question c

Compute the volume of \(B\).

Hint

The volume of \(B\) can be computed as the integal over the function \(1\) - and do not forget the absolute value of the Jacobian determinant as the integrand.

Answer

\(32\)

2: Mass Distributions in the \((x,y)\) plane

Consider the sets of points in \(\mathbb{R}^2\) given by:

\[\begin{equation*} B=\left\lbrace (x,y)\vert 1\leq x\leq 2 \, \wedge \, 0\leq y\leq x^3\right\rbrace \end{equation*}\]

and (again)

\[\begin{equation*} C=\left\lbrace (x,y)\Big\vert \frac 32\leq y \leq \frac 52 \wedge 0\leq x\leq \frac 12 y^2\right\rbrace. \end{equation*}\]

We interpret \(f(x,y)\) as the mass density (with units of \(\mathrm{kg/m^2}\)) at the point \((x,y)\).

Question a

Assume that the mass density is constant \(f(x,y)=1\) for \((x,y)\in B\). Determine the mass and the centre of mass of \(B\).

Hint

The mass is \(M = \int_B 1 \,\,\mathrm{d}(x,y)\). Find the formula for the centre of mass in the book in the section concerning integration of vector functions.

Answer

The mass is \(\frac{15}{4}\) and the centre of mass is \((x,y)=(\frac{124}{75},\frac{254}{105})\).

Question b

Assume that the mass density is \(f(x,y)=x^2\) for \((x,y)\in B\). Compute the mass as well as the centre of mass of \(B\).

Answer

The mass is \(\frac{21}{2}\) and the centre of mass is \((x,y)=(\frac{254}{147},\frac{73}{27})\).

Question c

Assume that the mass density is constant \(f(x,y)=1\) for \((x,y)\in C\). Compute the mass and the centre of mass of \(C\).

Answer

The mass is \(\frac{49}{24}\) and the centre of mass is \((x,y)=(\frac{4323}{3920},\frac{102}{49})\).

Question d

Assume that the mass density is \(f(x,y)=x^2\) for \((x,y)\in C\). Compute the mass and the centre of mass of \(C\).

Answer

The mass is \(\frac{37969}{10752}\) and the centre of mass is \((x,y)=(\frac{6767047}{3645024},\frac{84014}{37969})\).

3: Spherical Regions of 3D Space

Consider the spatial region \(\pmb{r}(\Gamma)\) given by

\[\begin{equation*} \pmb{r}(u,v,w)=\big(u\sin(v)\cos(w),u\sin(v)\sin(w),u\cos(v)\big), \quad (u,v,w) \in \Gamma, \end{equation*}\]

where \(\Gamma = [a,b] \times [c,d] \times [e,f] \subset [0, \infty[ \times [0,\pi] \times [0,2\pi]\). In other words, we are considering the following parameter values: \(u\in [a,b],v\in [c,d],w\in [e,f]\).

Question a

What do these parameters represent?

Question b

Let \(A\) be the region that we achieve with the choice:

\[\begin{equation*} a=1,b=3,c=\frac{\pi}{4},d=\frac{\pi}{3},e=0,f=\frac{3\pi}{4}, \end{equation*}\]

and \(B\) the region achieved with the choice:

\[\begin{equation*} a=2,b=4,c=\frac{\pi}{4},d=\frac{\pi}{2},e=-\frac{\pi}{4},f=\frac{\pi}{4}. \end{equation*}\]

Describe in words each of the regions \(A\), \(B\), and \(A\cap B\), and compute their volumes.

Answer

\(\mathrm{vol} (A)=\frac{13\pi(\sqrt 2-1)}{4}\)

\(\mathrm{vol} (B)=\frac{14\pi\sqrt 2}{3}\)

\(\mathrm{vol} (A\cap B)=\frac{19\pi(\sqrt 2-1)}{24}\)

By the way, \(\mathrm{vol}(A\cup B)\) can be computed using \(\mathrm{vol}(A)\) + \(\mathrm{vol}(B)\) - \(\mathrm{vol}(A\cap B)\).

Question c

Compute the integrals

\[\begin{equation*} \int_A x_1 \, \mathrm{d}\pmb{x}\,,\,\, \int_Bx_1 \, \mathrm{d}\pmb{x}\,,\,\text{ and } \int_{A\cap B}x_1 \, \mathrm{d}\pmb{x}. \end{equation*}\]

Answer

\[\begin{equation*} \int_A x_1 \, \mathrm{d}\pmb{x}=\frac{5\sqrt 2}{2}+\frac{5\pi\sqrt 2}{12}-\frac{5\sqrt 2\sqrt 3}{4} \end{equation*}\]

\[\begin{equation*} \int_B x_1 \, \mathrm{d}\pmb{x}=\frac{15\pi\sqrt 2}{2}+15\sqrt 2 \end{equation*}\]

\[\begin{equation*} \int_{A\cap B}x_1 \, \mathrm{d}\pmb{x}=\frac{65\sqrt 2}{32}+\frac{65\pi\sqrt 2}{192}-\frac{65\sqrt 2\sqrt 3}{64} \end{equation*}\]

4: An Indefinite Integral in the Plane

Let \(B\) be the unit square \([0,1]\times[0,1]\). We want to investigate the indefinite plane integral

\[\begin{equation*} I := \int_B \frac{1}{x_2-x_1-1} \mathrm{d}\pmb{x}. \end{equation*}\]

The integrand \(f(x_1,x_2)=\frac{1}{x_2-x_1-1}\) is not Riemann integrable over \(B\), since \(f\) is not defined at the point \((x_1,x_2)=(0,1)\). We wish to find out whether we can still find an integration value by assigning a limit value to the integral.

Question a

Find those points in the \((x,y)\) plane where \(f(x_1,x_2)\) is not defined. Find the image of \(f\) as a function on \(B \setminus \{(0,1)\}\).

Hint

Is \(f\) positive or negative on \(B\)?

Question b

Let \(B_a = [a,1] \times [0,1]\) for a fixed \(a \in [0,1]\). Draw a sketch of \(B_a\), and create a parametrization of \(B_a\). Compute the Jacobian determinant of your parametrization.

Question c

Compute the Riemann integral

\[\begin{equation*} I_a := \int_{B_a} \frac{1}{x_2-x_1-1} \,\mathrm{d}\pmb{x} \end{equation*}\]

for each \(a \in ]0,1]\).

Question d

Compute the limit value of \(I_a\) for \(a \to 0\).

Answer

\(I = -2 \ln(2)\)

Question e

Let \(B_b = [0,1] \times [0,b]\). Define \(I_b := \int_{B_b} \frac{1}{x_2-x_1-1} \,\mathrm{d}\pmb{x} \). Compute \(\lim_{b \to 1} I_b\) and compare with the above results.