# Multiple Integration

Business I · Mathematics II · https://tryals.app/learn/business-i/multiple-integration

## Multiple Integrals & Rectangles

A **double integral** sums a function over a 2D region, extending single-variable integration from intervals to areas:

$$\iint_R f(x,y)\, dA$$

**Iterated integration** evaluates this via nested single integrals. For a rectangular region $[a,b] \times [c,d]$:

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

| Property | Rule / Meaning |
| --- | --- |
| **Fubini's Theorem** | Order doesn't matter if $f$ is continuous; swap $dx$ and $dy$ freely. |
| **Integration Strategy** | Choose the order that makes the inner integral routine, not ugly. |

**Common pitfall:** Using constant limits for non-rectangular regions. If the region is a triangle below $y = x$, inner limits must be functions ($0$ to $x$). Copying rectangle limits integrates the wrong shape.

## Regions, Economics & Jacobian

For non-rectangular regions bounded by curves, the inner limits become functions:

$$\int_a^b \int_{h(x)}^{g(x)} f(x,y)\, dy\, dx$$

**Economic application**: In multi-good settings, **consumer surplus** and producer surplus extend from single integrals to double integrals over regions where willingness to pay exceeds market price.

**Change of variables**: Coordinate transformations use the **Jacobian** determinant absolute value $|J|$:

$$\iint f(x,y)\, dx\, dy = \iint f(x(u,v), y(u,v)) |J|\, du\, dv$$

| Tool | When to Use |
| --- | --- |
| **Polar Coordinates** | Essential when the integration region is naturally circular. |
| **Linear Substitution** | Used when a change of variables simplifies a difficult integrand. |

## Practice questions

6 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Fubini's theorem lets you swap the order of integration for any bounded function.

**Answer:** False

**Why:** False. It needs continuity on the region, or, in the Lebesgue form, absolute integrability. Without it the two iterated integrals can genuinely differ, which is the point of the classic counterexamples. In practice the hypothesis almost always holds, but it is a hypothesis rather than a licence.

Page: https://tryals.app/practice/business-i/multiple-integration/fubinis-theorem-lets-you-swap-the-order-of-integration-for-any

### 2. Estimate $\int_0^4 (8 - 2x)\, dx$, the area of the triangle with base 4 and height 8.

**Answer:** 16 (within ±2)

**Why:** $\int_0^4 (8-2x)\, dx = [8x - x^2]_0^4 = 32 - 16 = 16$. Or: triangle area $= \frac{1}{2} \times 4 \times 8 = 16$.

Page: https://tryals.app/practice/business-i/multiple-integration/estimate-0-8-2x-dx-the-area-of-the-triangle-with-base-4-and

### 3. Match each integration concept to its meaning.

**Answer:**

- Iterated integral → Evaluate a double integral as two nested single integrals
- Fubini's theorem → Order of integration can be switched for continuous functions
- Jacobian → Scaling factor for a change of variables in multiple integrals
- Consumer surplus → Area between demand curve and market price

**Why:** Iterated integrals compute the double integral; Fubini lets you swap order; the Jacobian handles coordinate changes; surplus is the economic payoff.

Page: https://tryals.app/practice/business-i/multiple-integration/match-each-integration-concept-to-its-meaning

### 4. Fubini's theorem lets you swap the order of integration over a rectangle. Why is that practically valuable rather than merely elegant?

A. Because it reduces the number of integrations from two to one
B. Because one order can be elementary and the other intractable
C. Because integrating in both orders and averaging improves precision
D. Because the two orders give different answers, and one must be chosen

**Answer:** B. Because one order can be elementary and the other intractable

**Why:** Both orders give the same number and one may be far easier to reach, the classic case being integrals with no elementary antiderivative in one variable and a trivial one in the other. Option 4 gets it backwards: equality of the two orders is precisely what the theorem asserts.

Page: https://tryals.app/practice/business-i/multiple-integration/fubinis-theorem-lets-you-swap-the-order-of-integration-over-a

### 5. Sort these into single integrals vs double integrals.

**Answer:**

- Single integral: $\int_0^1 x^2\, dx$, Area under a curve, Consumer surplus triangle
- Double integral: $\iint_R f(x,y)\, dA$, Volume under a surface, Joint surplus over a two-good region

**Why:** Single integrals compute areas; double integrals compute volumes or areas in higher-dimensional settings.

Page: https://tryals.app/practice/business-i/multiple-integration/sort-these-into-single-integrals-vs-double-integrals

### 6. Over the rectangle $[0,3]\times[0,2]$ the two orders of integration have the same limits; over a triangle they do not. What does Fubini actually guarantee?

A. That the region must be rectangular for the integral to exist
B. That the order of integration can never affect the limits
C. That the two iterated integrals agree in value, not in limits
D. That any double integral may be computed as a product of two single integrals

**Answer:** C. That the two iterated integrals agree in value, not in limits

**Why:** Equal values, different descriptions. Copying the old limits after a swap is the standard error, and it is invisible on a rectangle, which is why rectangles teach the wrong lesson.

Page: https://tryals.app/practice/business-i/multiple-integration/over-the-rectangle-0-3-0-2-the-two-orders-of-integration-have-the