Question

Let \(D\) be the solid bounded by \(y=x, z=1-y^{2}, x=0\) and \(z=0 .\) Write triple integrals over \(D\) in all six possible orders of integration.

Short answer

Question: Write down the triple integrals over solid D bounded by surfaces \(y=x, z=1-y^{2}, x=0\), and \(z=0\) in all six possible orders of integration. Answer: 1. $$\int_{0}^{1} \int_{x}^{1} \int_{0}^{1-y^2} dz\, dy \, dx$$ 2. $$\int_{0}^{1} \int_{0}^{x} \int_{0}^{1-y^2} dz\, dy \, dx$$ 3. $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{y} dx\, dz\, dy$$ 4. $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{y} dx\, dy\, dz$$ 5. $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{x} dy\, dx\, dz$$ 6. $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{x} dz\, dx\, dy$$

Step-by-step solution

Limits of Integration for dz, dy, dx order

First, we will determine the limits for the z-integral. Observing the region formed, the solid is bounded between the planes \(z=0\) and \(z=1-y^2\). So, the limits on the z-integral are from 0 to \(1-y^2\). Next, we need to find the limits for the y-integral. The region is bounded between \(y=x\) and \(y=1\). So, the limits for y-integral are from y=x to y=1. Finally, we need the limits for the x-integral. The solid is bounded between x=0 and x=1. So, the limits for the x-integral are from 0 to 1. The triple integral in the \(dz\, dy \, dx\) order is: $$\int_{0}^{1} \int_{x}^{1} \int_{0}^{1-y^2} dz\, dy \, dx$$ Next, we'll rearrange the order of integration.

Limits of Integration for dy, dz, dx order

For this order, the z-integral limits remain the same (0 to \(1-y^2\)). The y-integral limits will be from 0 to x, and the x-integral limits will be from 0 to 1. The triple integral in the \(dy\, dz\, dx\) order is: $$\int_{0}^{1} \int_{0}^{x} \int_{0}^{1-y^2} dz\, dy \, dx$$ Now, we'll rearrange the order of integration again.

Limits of Integration for dx, dz, dy order

For this order, the z-integral limits remain the same (0 to \(1-y^2\)). The x-integral limits will be from 0 to y, and the y-integral limits will be from 0 to 1. The triple integral in the \(dx\, dz\, dy\) order is: $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{y} dx\, dz\, dy$$ The remaining three orders of integration are:

Limits of Integration for dx, dy, dz order

$$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{y} dx\, dy\, dz$$

Limits of Integration for dy, dx, dz order

$$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{x} dy\, dx\, dz$$

Limits of Integration for dz, dx, dy order

$$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{x} dz\, dx\, dy$$ In summary, the triple integrals in all six possible orders of integration over the solid D are: $$\int_{0}^{1} \int_{x}^{1} \int_{0}^{1-y^2} dz\, dy \, dx$$ $$\int_{0}^{1} \int_{0}^{x} \int_{0}^{1-y^2} dz\, dy \, dx$$ $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{y} dx\, dz\, dy$$ $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{y} dx\, dy\, dz$$ $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{x} dy\, dx\, dz$$ $$\int_{0}^{1} \int_{0}^{1-y^2} \int_{0}^{x} dz\, dx\, dy$$

Key concepts

Multiple Integration

When tackling the complex topic of multiple integration, it is essential to start with a clear understanding of what it involves. The process of integrating a function of multiple variables over a region in space is known as multiple integration. Most commonly used in two or three dimensions, it helps in calculating volumes, masses, and other physical quantities that span across a given space.

Let's consider a practical example where multiple integration is applied. Imagine filling a swimming pool with water. If we want to calculate the amount of water needed, we would use double integration for a pool with uniform depth or triple integration if the depth varied throughout the pool. In the context of triple integrals, this process implies integrating a function in three steps, each corresponding to one of the three dimensions—typically denoted by x, y, and z.

The order of integration is flexible and can be changed based on the limits of integration and the region's shape. Remember, regardless of the order chosen, the final answer should be the same, indicating the total volume or mass of the region in question.

Integration Limits

Integration limits are essentially the boundaries that define the region over which you are integrating. When working with triple integrals, it's crucial to correctly determine these limits for each dimension. The limits can be constants, indicating a rectangle or box in space, or functions, outlining a more complex region such as a sphere or pyramid.

Understanding the importance of limits can be likened to setting the right boundaries for a game on a playground. Just as lines on the ground define where the game takes place, integration limits define where the calculation is applied in the space. These limits ensure that every point within the region is accounted for during integration and help avoid any overestimation or underestimation of the value of the integral.

Considering our original exercise, we've outlined a region 'D' that is constrained by several planes and curves. Each variable—z, y, and x—will have a corresponding set of limits that ensure we only integrate over region 'D'. Setting these limits carefully is quintessential to obtaining the correct evaluation of the triple integral.

Solid of Integration

The solid of integration refers to the specific three-dimensional region over which the integration is performed. This region is bounded by surfaces, which are represented by the equations provided in the problem statement. Think of it as a piece of rock that has certain distinctive boundaries; the goal is to find out certain properties about this whole 'chunk', like volume or mass.

In our exercise, the solid 'D' is uniquely determined by the planes and curves given by the equations, creating a closed bounded region in space. To properly visualize 'D', one might draw the region or use computer software to model it. This visual representation aids in setting up the correct limits for the integrals and understanding the interaction between the different surfaces that confine the solid.

Remember, identifying the solid of integration and its boundaries is critical before setting up the triple integral. Each surface or curve that forms the boundary of the solid corresponds to an integration limit. By understanding the solid's geometry, we are able to express the integral correctly in all possible orders of integration, assuring that our calculations accurately reflect the properties of the solid in question.