How to solve a Triple Integral problem?

How to solve a Triple Integral problem?

Triple integrals are a powerful mathematical tool that allows us to calculate the volume of three-dimensional regions. They have a wide range of applications in various fields, including physics, engineering, and economics. In this article, we will provide a comprehensive guide on how to solve triple integral problems step by step. From the basics of triple integrals to advanced techniques, we will cover everything you need to know to master this topic. So, whether you’re a student, a professional, or just someone interested in mathematics, this article is for you!

What is a Triple Integral?

A triple integral is an extension of a double integral, which is used to calculate the volume of a three-dimensional region. It is written as:

where R is the region of integration and f(x,y,z) is the function being integrated. The dV represents an infinitesimal volume element.

Properties of Triple Integrals

A triple integral is a useful tool in mathematics and science that has several important properties. These properties help us understand how triple integrals work and what they can do.

• 1. Linearity: A triple integral is linear, just like a regular integral. This means that if we add two functions together and then integrate them, we get the same result as if we integrate each function separately and then add them. In symbols, for functions f and g, the triple integral of a x f + b x g over a region R is equal to a times the triple integral of f over R plus b times the triple integral of g over R.
• 2. Boundaries: A triple integral is defined over a specific region R in three-dimensional space. This region has limits for the coordinates x, y, and z. We need to set up these limits correctly to calculate the triple integral accurately.
• 3. Change of Variables: Sometimes, we can make a triple integral easier by changing the variables. This is especially helpful when the region has a complicated shape. The change of variables formula requires us to find the Jacobian determinant, which measures how much the space changes due to the new variables.
• 4. Symmetry: A triple integral can be simplified if the region has symmetry. For example, if the region is the same on both sides of the xy-plane, the integral over that region might be zero because the positive and negative parts cancel out.
• 5. Order of Integration: When we integrate a function over a solid region, the order in which we integrate matters. Sometimes, changing the order can make the integral easier to solve. We should choose the order that matches the symmetry of the region.
• 6. Use in Calculus and Physics: Triple integrals are very useful in calculus, where they can help us find volumes, masses, and centers of mass for three-dimensional objects. In physics, they can help us calculate things like electric and gravitational potentials.

How to Solve a Triple Integral

To solve a triple integral, we need to follow these steps:

1. Identify the region of integration R.
2. Express the triple integral in terms of iterated integrals.
3. Evaluate the innermost integral first, then work your way outwards.

Let’s look at an example to see how this works.

Example 1

Calculate the volume of the region bounded by the planes x=0, y=0, z=0, and x+y+z=1

Solution

First, we need to identify the region of integration R. In this case, it is the tetrahedron bounded by the four planes mentioned above. Next, we express the triple integral as an iterated integral:

Now we evaluate the innermost integral first:

Substituting this into the next integral, we get:

Finally, we substitute this into the outermost integral:

Therefore, the volume of the region is 1661.

Sample Problems

Here are some additional sample problems for you to practice solving triple integrals:

Problem 1

Calculate the volume of the region bounded by the planes x=0, y=0, z=0, and x+2y+3z=6

Solution to Problem 1

Calculate the volume of the region bounded by the planes x=0,y=0,z=0, and x+2y+3z=6.

Solution

First, we need to identify the region of integration R. In this case, it is the tetrahedron bounded by the four planes mentioned above. Next, we express the triple integral as an iterated integral:

Now we evaluate the innermost integral first:

​

Substituting this into the next integral, we get:

Finally, we substitute this into the outermost integral:

Therefore, the volume of the region is 33.

Problem 2

Solution to Problem 2

Solution

First, we need to identify the region of integration �R. In this case, it is the solid bounded by the cylinder and the two planes mentioned above. Next, we express the triple integral as an iterated integral:

Now we evaluate the innermost integral first:

Substituting this into the next integral, we get:

Finally, we substitute this into the outermost integral:

Therefore, the volume of the region is 8π.

Problem 3

Solution to Problem 3

Solution

First, we need to identify the region of integration R. In this case, it is the tetrahedron in the first octant bounded by the coordinate planes and the plane x+y+z=4. Next, we express the triple integral as an iterated integral:

Now we evaluate the innermost integral first:

Substituting this into the next integral, we get:

Finally, we substitute this into the outermost integral:

Therefore, the value of the triple integral is 1616.

Problem 4

Solution to Problem 4

Let’s start with a quick sketch of Region E so we can get a feel for what we’re dealing with.

We’ve given the sketches with a set of “traditional” axes as well as a set of “box” axes to help visualize the surface and region. In this case, we’re dealing with the upper half of a sphere of radius 4. Now, since we are integrating over a portion of a sphere it makes sense to use spherical coordinates for the integral and the limits are,

Remember that φ is the angle from the positive z-axis that we rotate through as we cover the region and θ is the angle we rotate around the z-axis as we cover the region. In this case, we have the full upper half of the sphere so θ will range from 0 to 2π while φ will range from 0 to π/2 Plugging these limits into the integral and converting them to spherical coordinates gives,

Don’t forget to convert the x and z into spherical coordinates and also don’t forget that dV=ρ²sinφ dρdθdφ and so we’ll pick up a couple of extra terms when converting the dV to spherical coordinates. Okay, now all we need to do is evaluate the integral. Here is the ρ integration.

Next, let’s do the θ integration.

Finally, we’ll do the φ integration.

Note that, in this case, because the limits of each of the integrals were all constants we could have done the integration in any order we wanted to. In this case, it might have been “simpler” to do the φ first or second as that would have greatly reduced the integrand for the remaining integral(s).

Problem 5

Evaluate the following integral by first converting to an integral in spherical coordinates.

The Solution to Problem 5

First, let’s just get the Cartesian limits from the integral.

Now we need to convert the integral into spherical coordinates. Let’s first take care of the limits. From the upper z limit, we see that we are under z=√7−x²−y² (which is just the equation for the upper portion of a sphere of radius √7). From the lower z limit, we see that we are above z=√6x²+6y² (which is just the equation of a cone). So, we appear to be inside an “ice cream cone” shaped region as we usually are when dealing with spherical coordinates. This leads us to the following ρ limits.

0≤ρ≤√7

The limits for φ we can get them from the equation of the cone which is the lower z limit referenced in the previous step. First, we know that in terms of cylindrical coordinates, √x²+y²=r and we know that, in terms of spherical coordinates, r=ρsinφ. Therefore, if we convert the equation of the cone into spherical coordinates we get,

Because the region we are working on is above the cone we know that φ must therefore range from 0 to 0.3876.

Finally, let’s get the θ limits. For reference purposes here are the x and y limits we found in Step 1.

All the y limits tell us is that the region D from the original Cartesian coordinates integral is a portion of the circle of radius 1. Note that this should make sense as this is also the intersection of the sphere and cone we get from the z limits (we’ll leave it to you to verify this statement).

Now, from the x limits we see that we must have the left side of the circle of radius 1 and so the limits for θ are then,

​

The full set of spherical coordinate limits for the integral are then,

Okay, let’s convert the integral into spherical coordinates.

Don’t forget to convert the y into spherical coordinates. Also, don’t forget that the dzdydx comes from the dV in the original triple integral. We also know that, in terms of spherical coordinates,dV=ρ²sinφ dρdθdφ and so we in turn know that,

dzdydx=ρ²sinφ dρdθdφ Okay, now all we need to do is evaluate the integral. Here is the ρ integration.

Next, let’s do the θ integration.

So, as noted above once we got the integrand down to zero there was no reason to continue integrating as the answer will continue to be zero for the rest of the problem.

Don’t get excited about it when these kinds of things happen. They will on occasion and all it means is that we get to stop integrating a little sooner than we would have otherwise.

Here are some tricks and shortcuts that can help you solve triple integral problems more easily:

1. Symmetry: If the integrand and the region of integration are both symmetric concerning one or more coordinate planes, you can express the triple integral as an iterated integral over just one octant and multiply the result by the appropriate factor. For example, if the integrand and region of integration are both symmetric concerning all three coordinate planes, you can express the triple integral as an iterated integral over just one octant and multiply the result by 8.

2. Change of Variables: Sometimes, it can be helpful to change variables to make the region of integration or the integrand simpler. For example, if the part of integration is a sphere or a ball, it can be helpful to change to spherical coordinates. Similarly, if the region of integration is a cylinder or a disk, it can be beneficial to switch to cylindrical coordinates.

3. Integration by Parts: If the integrand is a product of two functions, you can sometimes use integration by parts to simplify the calculation. This involves expressing one of the functions as a derivative and integrating the other function.

4. Substitution: If the integrand contains a complicated expression, you can sometimes use substitution to simplify the calculation. This involves substituting a new variable for the complicated expression and adjusting the limits of integration accordingly.

FAQs (Frequently Asked Questions)

Q1: How is a triple integral different from three single integrals?

A: A triple integral integrates a function over a three-dimensional region, taking into account how the function changes in all three directions. Three single integrals integrate a function concerning one variable at a time, ignoring the other two.

Q2: How do I choose the order of integration for a triple integral?

A: The order of integration should follow the symmetry of the region, if any. We should start with the variable that has the smallest range in the region, and then move outward. We can use the shape of the region to guide us.

Q3: Can I use a triple integral for any shape of the region?

A: A triple integral can handle many shapes of regions, but some are more difficult than others. For complex shapes, we can use a change of variables to make them simpler. The new variables should fit well with the shape of the region.

Q4: How does symmetry affect the calculation of a triple integral?

A: Symmetry can make a triple integral easier by making some parts cancel out or become zero. If the region is symmetric about a plane, the integral over that plane might be zero.

Q5: Can I use numerical methods to approximate a triple integral?

A: Yes, if an exact solution is hard to find, we can use numerical methods like Monte Carlo simulation to estimate a triple integral with the desired accuracy.