Seen Questions

Double Riemann sum

Upper and lower double Riemann sums

Riemann integrability and an upper sum

Let $f:\mathbb R^2\to\mathbb R$ be a function defined on a rectangular region $D\subset\mathbb R^2$. Let $P$ be any partition of $D$ into subrectangles $\{R_{ij}\}$, let $(x_{ij}^*,y_{ij}^*)$ be points in $R_{ij}$, and let $L(f,P)$ and $U(f,P)$ denote the lower and upper sums of $f$ over $P$. Assume that $f$ is bounded on $D$.

(a)

Which of the following guarantees that $f$ is integrable over $D$? Select all answers that apply.

1. $\displaystyle \inf_P L(f,P)=\sup_P U(f,P)$.
2. For each $\varepsilon>0$ there exists a partition $P$ of $D$ such that $\displaystyle L(f,P)-U(f,P)<\varepsilon$.
3. For each $\varepsilon>0$ there exists a partition $P$ of $D$ such that $\displaystyle U(f,P)-L(f,P)<\varepsilon$.
4. For every sequence of tagged rectangular partitions $(P_m)$ of $D$ whose mesh tends to zero, the corresponding tagged sums converge to a common limit independently of the choice of tags.
5. $f(x,y)\equiv0$ on $\mathbb R^2\setminus D$.

(b)

Kenneth is considering the problem of approximating the double integral of a function using an upper Riemann sum. Let

Let $P_x$ be a partition of $[3,9]$ into $3$ equal parts and let $P_y$ be a partition of $[2,7]$ into $2$ equal parts. Let $P$ be the partition generated by the rectangular regions associated to $P_x\times P_y$.

Find the area $A$ of each subrectangle and compute the upper Riemann sum $U(f,P)$. Identify the corresponding boxes relative to the graph of $f$.

Solution

Part (a) tests two equivalent forms of the definition of integrability: upper and lower sums bound $f$ on each subrectangle, while tagged sums use arbitrary sample points. Part (b) uses the fact that $f$ is increasing on the rectangle, so its maximum on each subrectangle is at the upper-right corner.

(a) Selecting the integrability criteria

For a partition $P$, the lower and upper sums satisfy

The upper/lower-sum criterion says that $f$ is integrable when, for every $\varepsilon>0$, there is a partition for which

Therefore statement 3 is correct.

Statement 1 is a strong sufficient condition. If $c=\inf_P L(f,P)=\sup_P U(f,P)$, then for every partition $P$,

so $L(f,P)=U(f,P)=c$. The gap between upper and lower sums is zero, so $f$ is integrable. The condition is stronger than necessary, but it does guarantee integrability.

Statement 2 has the wrong sign: since $L(f,P)-U(f,P)\le0$ always, the inequality says nothing about the size of the gap. Statement 5 concerns values outside $D$, whereas integrability over $D$ depends only on $f$ on $D$.

Statement 4 is the tagged-sum form of the definition: the partitions become arbitrarily fine and the limit does not depend on the tags. Without that independence, different choices of tags could give different limits. Hence the correct selections are

(b) Computing the upper sum

The width of $R_1$ is $9-3=6$. Dividing it into $3$ equal parts gives

The height is $7-2=5$. Dividing it into $2$ equal parts gives

Thus the area of every subrectangle is

The partition points are

On $R_1$, both $x$ and $y$ are positive and

So $f$ is increasing in both variables and its maximum on each subrectangle is at the upper-right corner. The upper-right $x$-values are $5,7,9$ and the upper-right $y$-values are $\frac92,7$. Hence

This upper sum is the total volume of the boxes on the subrectangles of $P$ that reach above the graph. The lower sum would use the lower-left corners, giving boxes that fit below the graph, as in the double Riemann sum definition.

Polar area element

Changing variables in an elliptical integral

Your friend Brad is interested in calculating the following integral:

Here $\Omega$ is the region bounded by

(a) Choosing the change of variables

Brad wants to find a change of variables to write this integral in terms of coordinates that are very similar to the standard polar coordinates. Choose the suitable change of variables for the region $\Omega$ from the following options:

1. $x=4r\cos\theta,\ y=5r\sin\theta$ where $0\le\theta\le2\pi$ and $-1\le r\le1$.
2. $x=4r\cos\theta,\ y=5r\sin\theta$ where $0\le\theta\le\pi$ and $-1\le r\le1$.
3. $x=4r\cos\theta,\ y=5r\sin\theta$ where $-\pi\le\theta\le0$ and $0\le r\le1$.
4. $x=4r\cos\theta,\ y=5r\sin\theta$ where $0\le\theta\le2\pi$ and $0\le r\le1$.
5. $x=4r\cos\theta,\ y=5r\sin\theta$ where $0\le\theta\le\pi$ and $0\le r\le1$.
6. $x=4r\cos\theta,\ y=5r\sin\theta$ where $-\pi\le\theta\le0$ and $-1\le r\le1$.

The suitable option is 5:

To check the region, substitute the transformation into the equation of the curved boundary:

Thus $r=1$ gives the ellipse

whose upper half is the given curve. The condition $0\le r\le1$ fills the inside of the ellipse and $0\le\theta\le\pi$ gives $y=5r\sin\theta\ge0$.

The other angular ranges describe the lower half or the full ellipse. Allowing negative $r$ gives some points a second representation.

(b) Finding $g(r,\theta)$

With the order of integration requested in the question, the transformed integral has the form

The Jacobian matrix is

Therefore,

Since $0\le r\le1$, the absolute Jacobian factor is $20r$. The integrand becomes

Multiplying by the area factor gives

Here $a_1=0$, $a_2=1$, $\theta_1=0$ and $\theta_2=\pi$.

(c) Calculating $J$

Substituting the transformed integrand and bounds into (2) gives

The radial factor is

For the angular factor, expand the square:

Hence

Scaled polar coordinates suit this problem because they turn the upper half ellipse into the rectangle $0\le r\le1$, $0\le\theta\le\pi$, with the Jacobian supplying the area scaling.

The vanishing mixed term reflects the symmetry of the upper half ellipse under $x\mapsto-x$: the $\sin\theta\cos\theta$ term comes from the part of the integrand that cancels between the left and right halves.