GRAPHS

• Graph 1 — \(dA = dx \, dy\) — Finding the \(x\)-limits (inner bounds of integration).
• Graph 2 — \(dA = dx \, dy\) — Finding the \(y\)-limits (outer bounds of integration).
• Graph 3 — \(dA = dy \, dx\) — Finding the \(y\)-limits (inner bounds of integration).
• Graph 4 — \(dA = dy \, dx\) — Finding the \(x\)-limits (outer bounds of integration).
• Graph 5 — The planar region \(D\) bounded by the parabola \(x = y^2\) and the line \(x = y + 2\).

PROCESS — Finding Limits of Integration

Limits of integration of the double integral \(\iint_D f(x,y) \, dA\) correspond to the equations of the boundary curves of \(D\).

Order of Integration: \(dA = dx \, dy\)

Step 1 — Find the \(x\)-limits.

In your notebook:

1. Sketch the region of integration \(D\), label its boundary curves in the form \(x = g(y)\), and label the coordinates of any points where boundary curves intersect.
2. Draw an arrow parallel to the \(x\)-axis, in the direction of increasing \(x\)-values (pointing to the right).
3. The (inner) \(x\)-limits are the boundary curves of \(D\) where the arrow enters/leaves \(D\).

Step 2 — Find the \(y\)-limits.

Using the same sketch:

1. Project the region of integration \(D\) onto the \(y\)-axis.
2. The (outer) \(y\)-limits are the endpoints \(a \leq y \leq b\) of the projection.

Order of Integration: \(dA = dy \, dx\)

Step 1 — Find the \(y\)-limits.

In your notebook:

1. Sketch the region of integration \(D\), label its boundary curves in the form \(y = g(x)\), and label the coordinates of any points where boundary curves intersect.
2. Draw an arrow parallel to the \(y\)-axis, in the direction of increasing \(y\)-values (pointing up).
3. The (inner) \(y\)-limits are the boundary curves of \(D\) where the arrow enters/leaves \(D\).

Step 2 — Find the \(x\)-limits.

Using the same sketch:

1. Project the region of integration \(D\) onto the \(x\)-axis onto the \(x\)-axis.
2. The (outer) \(x\)-limits are the endpoints \(a \leq x \leq b\) of the projection.

EXERCISE 1 — Finding Limits of Integration

Assumptions

\(D\) is the planar region bounded by the parabola \(y = x^2\) and the line \(y = x + 2\).

Instructions

1. Find Limits Over \(D\) When \(dA = dy \, dx\)

Follow the process outlined at the top of this page to find the limits of integration for the double integral:

\[\iint_D f(x,y) \, dA\]

Using the order of integration \(dA = dy \, dx\).

2. Find Limits Over \(D\) When \(dA = dx \, dy\)

Follow the process outlined at the top of this page to find the limits of integration for the double integral:

\[\iint_D f(x,y) \, dA\]

Using the order of integration \(dA = dx \, dy\).

3. Compare

For this particular region, there is a substantial difference in the integrals produced by the two different orders of integration \(dA = dx \, dy\) vs \(dA = dy \, dx\). What is it?

EXERCISE 2 — CHALLENGE PROBLEM — Limits of Integration & Boundary Curves

Consider the integral:

\[\iint_D e^{y^2} \, dA = \int_{0}^{2} \int_{x/2}^{1} e^{y^2} \, dy \, dx\]

You'll show that reversing the order of integration results in an integral you can evaluate.

Instructions

1. Use existing limits to graph the region of integration \(D\).

Sketch the region \(D\) enclosed by the curves \(y = x/2\), \(y = 1\), and \(x = 0\).

These boundary curves correspond to limits of integration for the double integral:

\[\int_{0}^{2} \int_{x/2}^{1} e^{y^2} \, dy \, dx\].

2. Find limits to reverse the order of integration.

Use the process outlined at the top of this page, and your sketch of \(D\), to reverse the order of integration from \(dA = dy \, dx\) to \(dA = dx \, dy\).

3. Evaluate the new iterated integral.

Note that the original integral — \( \int_{0}^{2} \int_{x/2}^{1} e^{y^2} \, dy \, dx \) can't be evaluated exactly, because you can't find an explicit antiderivative for the function \( f(y) = e^{y^2} \).

But if you reverse the order of integration, you'll be able to evaluate using \(u\)-substitution.

Big Ideas — The Process

Trust the process.

Big Ideas — Limits of Integration & Boundary Curves

If the boundary curves of a region of integration are continuous, the order of integration does not affect the numeric value of a double integral.

In practice, there are additional considerations that may make one order preferred over the other.

Big Ideas — Reversing the Order of Integration

Limits of integration for a double integral \(\iint_D f(x,y) \, dA\) correspond to the boundary curves of the region of integration \(D\).