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Double Integrals on Rectangles · Tutorial 524 of 999

Common Errors in Rectangular Integral Bounds

Learn to match each pair of bounds to its variable, write both integration orders correctly, and check rectangular dimensions and differentials.

Beginner 9 min read

What You'll Learn

  • Match each pair of limits to the variable named by its differential.
  • Write the same rectangular domain in either integration order.
  • Recognize when reversed endpoints change the sign of an integral.
  • Distinguish an incomplete integral from a valid iterated integral.
  • Compute rectangle side lengths from endpoint differences.
  • Check an evaluated integral against direct bounds and area calculations.

Why Small Notational Errors Matter

A double integral over a rectangle is often straightforward to set up, but a small mistake in the limits can change the region being integrated over. Putting the right endpoints next to the wrong variable, for example, may produce a valid integral over the wrong rectangle. Other errors—such as omitting a differential or using an endpoint instead of a side length—can make an expression incomplete or give an incorrect value.

The key is to read an iterated integral from the differentials outward. The differential nearest the integrand identifies the inner variable; its limits give that variable's interval. The remaining differential identifies the outer variable and its limits. This simple check is useful both before and after evaluating an integral.

Definition: A rectangle with sides parallel to the coordinate axes can be written as \(R=[a,b]\times[c,d]\), where \(a\le b\) and \(c\le d\). Its \(x\)-coordinates range from \(a\) to \(b\), and its \(y\)-coordinates range from \(c\) to \(d\). Its side lengths are \(b-a\) and \(d-c\).

Read the Differentials Before the Limits

For a continuous function \(f\) on \(R=[a,b]\times[c,d]\), Fubini's theorem permits either integration order. In the first order, \(y\) is integrated first and \(x\) second:

$$ \iint_R f(x,y)\,dA = \int_a^b\int_c^d f(x,y)\,dy\,dx. $$

Here \(dy\) is adjacent to \(f(x,y)\), so the inner integral is with respect to \(y\), from \(c\) to \(d\). The outer integral is with respect to \(x\), from \(a\) to \(b\). The same rectangle in the other order is

$$ \iint_R f(x,y)\,dA = \int_c^d\int_a^b f(x,y)\,dx\,dy. $$

Now \(dx\) is the inner differential, so \(x\) runs from \(a\) to \(b\), while the outer variable \(y\) runs from \(c\) to \(d\). The limits move with their coordinate intervals when the order is reversed; they do not stay in the same positions in the notation.

Rule: In \(\int_A^B\int_C^D f(x,y)\,dy\,dx\), \(C,D\) are the \(y\)-limits and \(A,B\) are the \(x\)-limits. In \(\int_C^D\int_A^B f(x,y)\,dx\,dy\), \(A,B\) are still the \(x\)-limits and \(C,D\) are still the \(y\)-limits.

Worked Example: Checking Both Integration Orders

Worked Example: Checking Both Integration Orders

Evaluate \(\iint_R xy\,dA\) for \(R=[1,3]\times[0,2]\) in both orders and confirm that the bounds match the rectangle.

For the order \(dy\,dx\), \(y\) runs from \(0\) to \(2\), and \(x\) runs from \(1\) to \(3\):

$$ \begin{aligned} \int_1^3\int_0^2 xy\,dy\,dx &=\int_1^3 x\left[\frac{y^2}{2}\right]_0^2\,dx\\ &=\int_1^3 x\left(\frac{4}{2}-0\right)\,dx\\ &=\int_1^3 2x\,dx\\ &=[x^2]_1^3\\ &=9-1=8. \end{aligned} $$

For the order \(dx\,dy\), \(x\) runs from \(1\) to \(3\), and \(y\) runs from \(0\) to \(2\):

$$ \begin{aligned} \int_0^2\int_1^3 xy\,dx\,dy &=\int_0^2 y\left[\frac{x^2}{2}\right]_1^3\,dy\\ &=\int_0^2 y\left(\frac{9}{2}-\frac{1}{2}\right)\,dy\\ &=\int_0^2 4y\,dy\\ &=[2y^2]_0^2\\ &=8-0=8. \end{aligned} $$

Both values are \(8\), as expected for the two orders of integration over the same rectangle. The bounds can also be checked directly: the \(x\)-interval is \([1,3]\) and the \(y\)-interval is \([0,2]\) in both expressions.

Keep Each Endpoint With Its Coordinate

A frequent error is to copy the \(x\)-limits into the inner integral after changing the order, or to assign the \(y\)-limits to \(x\). A useful safeguard is to write the rectangle first as \(R=[a,b]\times[c,d]\). This records the coordinate intervals separately: \([a,b]\) belongs to \(x\), and \([c,d]\) belongs to \(y\). Only then write the nested integral and place each interval according to the differentials.

Worked Example: Correcting Limits Assigned to the Wrong Variable

Evaluate the integral of \(x\) over \(R=[-1,2]\times[2,5]\). A common incorrect setup is \(\int_{-1}^2\int_2^5 x\,dx\,dy\): it puts the \(y\)-limits on the inner \(x\)-integral and the \(x\)-limits on the outer \(y\)-integral.

The rectangle specifies \(x\in[-1,2]\) and \(y\in[2,5]\). In the order \(dy\,dx\), the correct setup is

$$ \begin{aligned} \int_{-1}^2\int_2^5 x\,dy\,dx &=\int_{-1}^2 x[y]_2^5\,dx\\ &=\int_{-1}^2 x(5-2)\,dx\\ &=3\left[\frac{x^2}{2}\right]_{-1}^2\\ &=3\left(2-\frac{1}{2}\right)\\ &=\frac{9}{2}. \end{aligned} $$

The reversed order uses \(x\in[-1,2]\) for the inner integral and \(y\in[2,5]\) for the outer integral:

$$ \begin{aligned} \int_2^5\int_{-1}^2 x\,dx\,dy &=\int_2^5 \left[\frac{x^2}{2}\right]_{-1}^2\,dy\\ &=\int_2^5 \left(2-\frac{1}{2}\right)\,dy\\ &=\frac{3}{2}[y]_2^5\\ &=\frac{3}{2}(5-2)\\ &=\frac{9}{2}. \end{aligned} $$

The incorrect setup instead integrates \(x\) from \(2\) to \(5\), and \(y\) from \(-1\) to \(2\). Its value is \(\int_{-1}^2\int_2^5 x\,dx\,dy=\int_{-1}^2(25/2-4/2)\,dy=(21/2)(3)=63/2\). The calculation is possible, but it describes a different rectangle. It is not a harmless rearrangement of the original bounds.

Endpoints, Orientation, and Side Lengths

For an ordinary description of a rectangle, the lower endpoint should be written before the upper endpoint in each coordinate: \(a\le b\) and \(c\le d\). If a pair of limits is deliberately reversed, the definite integral changes sign in that variable. It does not describe the same positively oriented rectangular integral.

Side lengths are differences of endpoints, not the endpoints themselves. A rectangle spanning \(x=a\) to \(x=b\) has horizontal side length \(b-a\), whether or not either endpoint is zero. Similarly, its vertical side length is \(d-c\). In particular, an interval such as \([-1,3]\) has length \(3-(-1)=4\), not \(3\).

Check: For the constant function \(1\), the integral over \(R=[a,b]\times[c,d]\) equals the rectangle's area, \((b-a)(d-c)\). For a constant function \(k\), the value is \(k(b-a)(d-c)\).

This area check follows directly from evaluating the iterated integral. With \(a\le b\) and \(c\le d\),

$$ \begin{aligned} \int_a^b\int_c^d 1\,dy\,dx &=\int_a^b [y]_c^d\,dx\\ &=\int_a^b(d-c)\,dx\\ &=(d-c)[x]_a^b\\ &=(d-c)(b-a). \end{aligned} $$

The result is the product of the two side lengths. This also explains why using an endpoint in place of a side length can make a constant-function check fail.

Worked Example: Finding the Correct Side Lengths

Evaluate \(\iint_R 2\,dA\) on \(R=[-1,3]\times[2,5]\). The side lengths are \(3-(-1)=4\) and \(5-2=3\). Thus the area is \(4\cdot3=12\), and a constant height of \(2\) should give an integral of \(2\cdot12=24\).

Direct integration confirms the result:

$$ \begin{aligned} \int_{-1}^3\int_2^5 2\,dy\,dx &=\int_{-1}^3 2[y]_2^5\,dx\\ &=\int_{-1}^3 2(5-2)\,dx\\ &=\int_{-1}^3 6\,dx\\ &=6[x]_{-1}^3\\ &=6(3-(-1))\\ &=24. \end{aligned} $$

A mistaken side-length calculation might use \(3\) and \(5\), the upper endpoints, and conclude that the integral is \(2(3)(5)=30\). The endpoints are not the side lengths: the intervals start at \(-1\) and \(2\), so their lengths are \(4\) and \(3\). The direct evaluation and the area check both give \(24\).

Every Iterated Integral Needs Both Differentials

A standard iterated integral records two integrations, so it needs two differentials in the correct order. For example, \(\int_a^b\int_c^d f(x,y)\,dy\,dx\) says first to integrate with respect to \(y\), then with respect to \(x\). Writing \(\int_a^b\int_c^d f(x,y)\,dy\) leaves the outer integration unspecified. It is incomplete notation, not a complete double integral with a value.

Do not confuse an omitted differential with a differential that appears in the wrong place. In \(\int_a^b\int_c^d f(x,y)\,dx\,dy\), the notation is complete, but it identifies \(x\) as the inner variable and \(y\) as the outer variable. Its limits must therefore match those coordinate intervals.

Worked Example: Restoring an Omitted Differential

Evaluate the integral of \(x+y\) on \(R=[0,2]\times[1,3]\). An expression ending in \(dy\) after the inner integral, such as \(\int_0^2\int_1^3(x+y)\,dy\), is incomplete because it does not specify the outer integration with respect to \(x\). The complete setup is

$$ \begin{aligned} \int_0^2\int_1^3(x+y)\,dy\,dx &=\int_0^2\left[xy+\frac{y^2}{2}\right]_1^3\,dx\\ &=\int_0^2\left(3x+\frac{9}{2}-x-\frac{1}{2}\right)\,dx\\ &=\int_0^2(2x+4)\,dx\\ &=[x^2+4x]_0^2\\ &=(4+8)-0\\ &=12. \end{aligned} $$

For a check in the other order, \(x\) runs from \(0\) to \(2\) inside and \(y\) runs from \(1\) to \(3\) outside:

$$ \begin{aligned} \int_1^3\int_0^2(x+y)\,dx\,dy &=\int_1^3\left[\frac{x^2}{2}+yx\right]_0^2\,dy\\ &=\int_1^3(2+2y)\,dy\\ &=[2y+y^2]_1^3\\ &=(6+9)-(2+1)\\ &=12. \end{aligned} $$

The two complete iterated integrals agree, and both use the coordinate intervals specified by the rectangle.

A Short Diagnostic Routine

Before evaluating a rectangular double integral, check its structure in order. These checks do not replace the integration, but they can catch a setup error before it affects the computation.

1
Record the rectangle.
Write \(R=[a,b]\times[c,d]\), so the \(x\)-interval and \(y\)-interval are explicit.
2
Read the differentials.
The differential beside the integrand names the inner variable; the final differential names the outer variable.
3
Match limits to coordinates.
Use \([a,b]\) for \(x\) and \([c,d]\) for \(y\), regardless of which variable is integrated first.
4
Check interval lengths and orientation.
Compute \(b-a\) and \(d-c\), and confirm that the endpoints are in increasing order for the stated rectangle.
5
Check the result when possible.
For a constant integrand, compare with the constant times the rectangle's area. When reversing order, compare the two evaluated integrals.

Why These Checks Are Useful

Correctly identifying the bounds is part of defining the integral, not merely a matter of notation. A mistaken coordinate assignment can produce a perfectly evaluable expression for a different region, as the example with \(x\) over \([-1,2]\times[2,5]\) showed. Missing differentials leave the calculation unfinished. Incorrect side lengths can produce a result that conflicts with the basic area of the rectangle.

For continuous \(f\) on a rectangle, Fubini's theorem guarantees that the two correctly written integration orders give the same double integral. That agreement is useful as a check, but it cannot repair mismatched limits: both orders must first describe the same rectangle. A reliable habit is therefore to name the coordinate intervals explicitly before writing either nested integral.

Check Your Understanding

Use the rectangle and differentials in each question to identify the correct bounds or diagnose the error.

  1. For \(R=[-2,1]\times[3,4]\), write the iterated integral in the order \(dy\,dx\).
  2. For the same rectangle, write the iterated integral in the order \(dx\,dy\). Which variable is integrated first?
  3. What are the side lengths and area of \([1,5]\times[-3,2]\)?
  4. Why is \(\int_0^2\int_1^3 f(x,y)\,dy\) incomplete as a double integral?
  5. For \(R=[0,2]\times[1,4]\), what value should \(\iint_R 3\,dA\) have? State the side lengths used.

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