# Triple Integrals

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

## Triple Integrals Setup

A **Triple Integral** extends integration to three dimensions: $\iiint_E f(x,y,z)\,dV$. It evaluates from the inside out using iterated integrals:

$$\int_a^b\int_{g_1}^{g_2}\int_{h_1}^{h_2} f\,dz\,dy\,dx$$

To **Set Up Limits**, follow these three steps carefully:

| Step | Action | Purpose |
| --- | --- | --- |
| 1 | Sketch the region | Visualize $E$ |
| 2 | Project onto a plane | Find outer limits |
| 3 | Find the $z$-range | Set inner limits |

## Applications & Pitfalls

Integrals compute physical properties. **Volume** is $\iiint_E 1\,dV$. **Mass** is $M = \iiint_E \rho\,dV$. **Moment of inertia** about the $z$-axis is $I_z = \iiint_E (x^2+y^2)\rho\,dV$.

**Worked Example**: For a sphere $x^{2}+y^{2}+z^{2} \leq R^{2}$, the volume is $V = \frac{4}{3}\pi R^{3}$.

**Common Pitfall**: Reversing the order of integration is never just swapping limits. You must re-describe the region from scratch using your picture, not the algebra.

## Practice questions

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

### 1. $\int_{0}$¹ $\int_{0}$¹ $\int_{0}^{1} xyz dz$ dy dx = ? **Set the slider to your answer.**

**Answer:** 0.125 (within ±0.004)

**Why:** = ($\int_{0}^{1} x dx$)($\int_{0}^{1} y dy$)($\int_{0}^{1} z dz$) = ($\frac{1}{2}$)($\frac{1}{2}$)($\frac{1}{2}$) = 1/8 = 0.125.

Page: https://tryals.app/practice/physics-i/triple-integrals/0-0-0-xyz-dz-dy-dx-set-the-slider-to-your-answer

### 2. Match each solid to the natural coordinate system for its triple integral.

**Answer:**

- Rectangular box → Cartesian (x, y, z)
- Cylinder → Cylindrical (r, $\theta$, z)
- Sphere → Spherical ($\rho$, $\theta$, $\varphi$)
- Cone → Cylindrical or Spherical

**Why:** Matching coordinates to geometry simplifies the limits and integrand.

Page: https://tryals.app/practice/physics-i/triple-integrals/match-each-solid-to-the-natural-coordinate-system-for-its-triple

### 3. Which quantities can be computed using triple integrals?

A. Moment of inertia
B. Mass of a solid with varying density
C. Slope of a tangent line
D. Volume of a 3D region

**Answer:** A. Moment of inertia; B. Mass of a solid with varying density; D. Volume of a 3D region

**Why:** Triple integrals compute volume, mass, center of mass, and moments.

Page: https://tryals.app/practice/physics-i/triple-integrals/which-quantities-can-be-computed-using-triple-integrals

### 4. Match each physical quantity to the triple integral that computes it (over a solid $E$ of density $\rho(x,y,z)$).

**Answer:**

- Volume of $E$ → $\iiint_E 1\,dV$
- Mass of $E$ → $\iiint_E \rho\,dV$
- Average density → $\iiint_E \rho\,dV \;/\; \iiint_E 1\,dV$
- Moment about the $xy$-plane → $\iiint_E z\rho\,dV$

**Why:** One machine, many outputs: change the integrand and $\iiint dV$ measures volume (1), mass ($\rho$), or leverage ($z\rho$, the ingredient of centre-of-mass). Physics quantities are mostly "integrate the right weighting over the body".

Page: https://tryals.app/practice/physics-i/triple-integrals/match-each-physical-quantity-to-the-triple-integral-that-computes-it

### 5. To integrate over a **ball** of radius $R$ centred at the origin, which coordinate system makes the limits simplest, and why?

A. Any system, limits have the same complexity everywhere
B. Spherical: the ball becomes a box in $\rho,\varphi,\theta$
C. Cylindrical: the ball becomes a perfect cylinder
D. Cartesian: three symmetric square-root limits

**Answer:** B. Spherical: the ball becomes a box in $\rho,\varphi,\theta$

**Why:** Match the coordinates to the region’s symmetry and curved boundaries flatten into constant limits: the ball is literally a rectangular box in $(\rho, \varphi, \theta)$. The price, the volume factor $\rho^{2}\sin\varphi$, is far cheaper than nested square roots.

Page: https://tryals.app/practice/physics-i/triple-integrals/to-integrate-over-a-ball-of-radius-r-centred-at-the-origin-which

### 6. ∭_E 1 dV gives the volume of the region E.

**Answer:** True

**Why:** Integrating the constant 1 over a region gives its volume (or area in 2D).

Page: https://tryals.app/practice/physics-i/triple-integrals/e-1-dv-gives-the-volume-of-the-region-e