# Double Integrals

Physics I · Multivariable Calculus · https://tryals.app/learn/physics-i/double-integrals

## Double Integrals & Fubini

A **double integral** computes the **signed volume** under a surface $z = f(x,y)$ over a region $R$: $\iint_R f(x,y)\,dA$.

To evaluate, use **iterated integrals** by integrating one variable at a time:

$$\iint_R f\,dA = \int_a^b\int_{g_1(x)}^{g_2(x)} f\,dy\,dx$$

**Fubini's theorem** states that if $f$ is continuous on $R$, you can integrate in either order. Sketch $R$ first to choose the simpler order.

| Concept | Formula | Meaning |
| --- | --- | --- |
| **Area** | $\iint_R 1\,dA$ | Area of region $R$ |
| **Mass** | $\iint_R \rho\,dA$ | Total mass with density $\rho$ |

> **Common pitfall:** The **inner limits may depend on the outer variable, never the reverse**. Writing $\int_0^y \dots$ as the outer integral is a classic setup error.

## Applications & Center of Mass

Double integrals extend single-variable calculus to find average values and physical centers.

| Application | Formula | Description |
| --- | --- | --- |
| **Average Value** | $\bar{f} = \frac{1}{\text{Area}(R)}\iint_R f\,dA$ | Mean height of surface |
| **Center of Mass** | $\bar{x} = \frac{1}{M}\iint_R x\rho\,dA$ | Balance point coordinates |

**Worked insight:** Always sketch the region $R$ to set correct limits. The key skill is setting up boundaries properly before integrating.

> **Key takeaway:** If an integral looks impossible in one order, swap the limits using Fubini's theorem. Inner limits are always functions of the outer variable.

## Practice questions

5 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Rain falls with varying intensity $r(x,y)$ over a city. What does $\iint_{city} r(x,y)\,dA$ compute?

A. The area of the city
B. The peak rainfall intensity
C. The average rainfall intensity
D. The total volume per unit time

**Answer:** D. The total volume per unit time

**Why:** A double integral is a weighted total: local rate × patch area, summed to the whole. Divide by $\iint dA$ and you get the average intensity, total and average are the two faces of every double integral.

Page: https://tryals.app/practice/physics-i/double-integrals/rain-falls-with-varying-intensity-r-x-y-over-a-city-what-does

### 2. Swapping the order of integration in a double integral often makes it far easier. Why can the two integrals not simply share the same limits?

A. The integrand, which must be rewritten in the new order
B. The LIMITS, since they describe the region by the outer variable
C. The sign, which flips whenever the order is exchanged
D. Nothing: the limits and integrand both carry over unchanged

**Answer:** B. The LIMITS, since they describe the region by the outer variable

**Why:** The region is the same; its description is not. Sketching it is the reliable way to get the new limits, and Fubini guarantees the answers agree once they are right.

Page: https://tryals.app/practice/physics-i/double-integrals/swapping-the-order-of-integration-in-a-double-integral-often-makes-it

### 3. A rectangular plate $0 \le x \le 3$, $0 \le y \le 2$ has uniform density 5 kg/m². Using $\iint 5\,dA$, find its mass in kg.

**Answer:** 30 (within ±0.1)

**Why:** Mass $= 5 \times 6 = 30$ kg. Constant integrands turn integrals into multiplication, always the first sanity check before attempting harder iterated integrals: does the answer reduce to base × height reasoning?

Page: https://tryals.app/practice/physics-i/double-integrals/a-rectangular-plate-0-x-3-0-y-2-has-uniform-density-5-kg-m

### 4. Order the steps for evaluating $\iint_R xy\,dA$ over a triangle by **iterated integration**.

**Answer:**

1. Sketch the region and read off its boundary curves
2. Choose an order, e.g. $x$ inside from 0 to $y$, then $y$ outside from 0 to 1
3. Integrate the inner variable, treating the outer one as constant
4. Substitute the inner limits, leaving a function of the outer variable
5. Integrate the outer variable over its fixed range

**Why:** Iterated integration slices a 2D problem into 1D strips: the inner integral sums one strip; the outer integral sums the strips. Getting the region’s limits right (from the sketch!) is where every double-integral battle is won or lost.

Page: https://tryals.app/practice/physics-i/double-integrals/order-the-steps-for-evaluating-r-xy-da-over-a-triangle-by-iterated

### 5. Sort each application into what a double integral computes.

**Answer:**

- Area: ∬_R 1 dA, ∬_R dA over region R
- Volume: ∬_R $f(x,y)$ dA where $f \geq 0$ is height
- Mass: ∬_R $\rho$(x,y) dA ($\rho$ = density)

**Why:** Double integrals compute area (integrand 1), volume (integrand = height), or mass (integrand = density).

Page: https://tryals.app/practice/physics-i/double-integrals/sort-each-application-into-what-a-double-integral-computes