# Triple Integral, already solved, need checked

• King Tony
In summary, the region S in the first octant under the plane 3x + 2y + z = 4 has a volume of 448/27. The integral used to find this volume is \int^{\frac{4}{3}}_{0}\int^{\frac{3}{2}x + 2}_{0}\int^{-3x - 2y + 4}_{0}dzdydx, with the correct bounds being 0\leq{x}\leq\frac{4}{3}, 0\leq{y}\leq{2}-\frac{3}{2}x.
King Tony

## Homework Statement

Let S be the region in the first octant under the plane 3x + 2y +z = 4. Find the volume of S.

idk?

## The Attempt at a Solution

$$\int^{\frac{4}{3}}_{0}\int^{\frac{3}{2}x + 2}_{0}\int^{-3x - 2y + 4}_{0}dzdydx$$

= $$\int^{\frac{4}{3}}_{0}\int^{\frac{3}{2}x + 2}_{0}(-3x - 2y + 4)dydx$$

= $$\int^{\frac{4}{3}}_{0}(-\frac{15}{2}x^{2} - 6x + 4)dx$$

= 448/27

Last edited:
Sorry, took a little while to figure out the symbols and dealies, pretty sure it's good to go right now.

Mumble, mumble..gotta check the bounds:

The upper line in the z=0 plane obeys the equation 3x+2y=4

Thus,
$$0\leq{x}\leq\frac{4}{3}, 0\leq{y}\leq{2}-\frac{3}{2}x$$
Thus, you've got a sign wrong in the y-bound, according to my view.

arildno said:
Mumble, mumble..gotta check the bounds:

The upper line in the z=0 plane obeys the equation 3x+2y=4

Thus,
$$0\leq{x}\leq\frac{4}{3}, 0\leq{y}\leq{2}-\frac{3}{2}x$$
Thus, you've got a sign wrong in the y-bound, according to my view.

You're so right, thankyou!

## 1. What is a triple integral and how is it different from a regular integral?

A triple integral is a mathematical tool used to find the volume of a three-dimensional object. It is different from a regular integral in that it involves integrating over three variables instead of just one.

## 2. How do you solve a triple integral?

To solve a triple integral, you first need to set up the limits of integration for each variable. Then, you can use the appropriate integration techniques (such as substitution or integration by parts) to evaluate the integral.

## 3. Can you explain the concept of "slicing" in triple integrals?

Slicing is a technique used in triple integrals to break down a three-dimensional object into smaller, two-dimensional slices. Each slice is then integrated over to find the volume of that slice, and the results are added together to get the total volume of the object.

## 4. What are some real-life applications of triple integrals?

Triple integrals are commonly used in physics and engineering to calculate volumes, mass, and other physical properties of three-dimensional objects. They are also used in probability and statistics to calculate probabilities of events in three-dimensional space.

## 5. How can I check my solution for a triple integral?

One way to check your solution for a triple integral is to use graphing software to visualize the three-dimensional object and see if your calculated volume matches the actual volume. You can also try using different integration techniques to see if they result in the same answer.

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