Calculating Volumes: A Step-by-Step Guide

Jan 15, 2026 by Tim Redaksi 42 views

Hey guys! Let's dive into some calculus fun and figure out how to calculate the volumes of objects that sit under certain planes. We'll break down two specific problems, making sure we understand the concepts clearly. So, grab your pencils, and let's get started!

Part A: Volume Calculation with a Plane and Coordinate Planes

Alright, first things first, we're dealing with the plane defined by the equation 3x + 2y + z = 12. We want to find the volume of the region beneath this plane and bounded by a few other planes: x = 2, y = 3, and the three coordinate planes – that's the x-y plane (z=0), the x-z plane (y=0), and the y-z plane (x=0). Sounds like a lot, but don't worry; we'll break it down.

The core idea here is to use a triple integral. Basically, we're slicing up our 3D region into tiny, tiny rectangular boxes and summing up their volumes. The general setup for a volume calculation using a triple integral is: Volume = ∫∫∫ dV. Here, dV represents a tiny volume element. In Cartesian coordinates, we can write dV as dx dy dz. So, our integral will look something like this: Volume = ∫∫∫ dz dy dx. The order of integration (dx, dy, dz) does not change the answer, but it can affect the complexity of the calculation.

Before we jump into the integral, let's establish the limits of integration. These limits define the boundaries of our region. Let's tackle it step by step:

z limits: The plane 3x + 2y + z = 12 gives us the upper bound for z. Solving for z, we get z = 12 - 3x - 2y. The lower bound for z is the x-y plane, where z = 0. So, our z limits are from 0 to 12 - 3x - 2y.
y limits: We're given that y = 3 is one boundary. The other boundary for y is the y-z plane, where y = 0. So, our y limits are from 0 to 3.
x limits: We're given that x = 2 is one boundary. The other boundary for x is the x-z plane, where x = 0. So, our x limits are from 0 to 2.

Now we can set up our triple integral: Volume = ∫₀² ∫₀³ ∫₀¹²⁻³ˣ⁻²ʸ dz dy dx.

Time to solve this bad boy! We'll integrate step by step.

First, integrate with respect to z: ∫₀¹²⁻³ˣ⁻²ʸ dz = [z]₀¹²⁻³ˣ⁻²ʸ = (12 - 3x - 2y) - 0 = 12 - 3x - 2y.
Next, integrate with respect to y: ∫₀³ (12 - 3x - 2y) dy = [12y - 3xy - y²]₀³ = (36 - 9x - 9) - 0 = 27 - 9x.
Finally, integrate with respect to x: ∫₀² (27 - 9x) dx = [27x - (9/2)x²]₀² = (54 - 18) - 0 = 36.

So, the volume of the region is 36 cubic units. Pretty neat, huh? We’ve successfully used a triple integral to find the volume of a region bounded by a plane and several other planes. Great job!

This is a fundamental example of how we use calculus to solve real-world problems. Keep in mind that understanding the limits of integration is absolutely key!

Remember to draw a sketch of the region if you're ever stuck. Visualizing the problem can make all the difference, and a sketch can help you see those integration limits clearly.

Now, let's move onto the next part!

Part B: Volume Calculation with a Paraboloid and Coordinate Planes

Okay, guys, time to tackle another fun problem! This time, we are looking at the volume under the surface given by 4z = 16 - 2x² - y². This is a paraboloid. We're also limited by the planes x = 2, y = 4, and, like before, the coordinate planes.

Again, we'll use a triple integral to calculate the volume. The set up is much the same. However, the surface equation is different and will affect our limits of integration.

First, let's rewrite our surface equation to isolate z: z = (16 - 2x² - y²)/4 = 4 - (1/2)x² - (1/4)y². This is the upper bound for z. The lower bound is the x-y plane where z = 0.

Now let's find the limits for each variable:

z limits: The lower limit is z = 0 (x-y plane), and the upper limit is z = 4 - (1/2)x² - (1/4)y².
y limits: We're given that y = 4 is one boundary, and the y-z plane gives us y = 0. So, our y limits are from 0 to 4.
x limits: We're given that x = 2 is one boundary, and the x-z plane tells us that x = 0. Our x limits are from 0 to 2.

Now we can write out the triple integral: Volume = ∫₀² ∫₀⁴ ∫₀⁴⁻(¹/₂)x²⁻(¹/₄)y² dz dy dx.

Let's get this integral done! Let's solve it step by step:

First, integrate with respect to z: ∫₀⁴⁻(¹/₂)x²⁻(¹/₄)y² dz = [z]₀⁴⁻(¹/₂)x²⁻(¹/₄)y² = 4 - (1/2)x² - (1/4)y².
Next, integrate with respect to y: ∫₀⁴ (4 - (1/2)x² - (1/4)y²) dy = [4y - (1/2)x²y - (1/12)y³]₀⁴ = [16 - 2x² - (64/12)] - 0 = 16 - 2x² - (16/3) = (32/3) - 2x².
Finally, integrate with respect to x: ∫₀² ((32/3) - 2x²) dx = [(32/3)x - (2/3)x³]₀² = [(64/3) - (16/3)] - 0 = 48/3 = 16.

So the volume of this region is 16 cubic units.

See? Using triple integrals, we successfully determined the volume under a paraboloid, all the while respecting the boundary planes. Now you have two solid examples to help you understand volume calculations.

Conclusion: Mastering Volume Calculations

Alright, folks, we've successfully navigated through two volume calculation problems using triple integrals! The key takeaways are:

Understanding the setup: Know that you'll be using triple integrals to find these volumes, and that Volume = ∫∫∫ dV. Remember that dV can be written as dx dy dz, dy dz dx, and so on. The order matters for the complexity but not the result.
Determining the limits of integration: This is critical! Always start by figuring out the upper and lower bounds for each variable (x, y, and z) within the region. Sketching the region or imagining it can really help!
Integrating step by step: Integrate with respect to one variable at a time, working your way through the integral. Take your time, and double-check your calculations.

Don't worry if it feels a little tricky at first. It takes practice. The more problems you solve, the more comfortable you'll become with this. Keep practicing, and always remember to visualize the problem! If you're comfortable with double integrals, triple integrals are just the next logical step.

Keep in mind that finding volumes is a cornerstone of calculus, and it has applications in a lot of fields like engineering, physics, and computer graphics. Every problem is an opportunity to learn and hone your skills! You got this!

Great job sticking around and working through this with me. Keep exploring and happy calculating!