https://r2.erweima.ai/mathbot/image/1726440144884igiqqhen.jpg

The given image contains a triple integral of the form:

\[
\int_0^1 \int_0^2 \int_0^{\frac{1}{2}y} (x + y - z) \, dz \, dy \, dx
\]

To solve this, we will evaluate the integrals step by step, starting from the innermost integral to the outermost.

### Step 1: Evaluate the innermost integral with respect to \( z \)

\[
I_1 = \int_0^{\frac{1}{2}y} (x + y - z) \, dz
\]

The antiderivative of \( (x + y - z) \) with respect to \( z \) is:

\[
x + y - \frac{z^2}{2}
\]

Now, substitute the limits \( z = 0 \) and \( z = \frac{1}{2}y \):

\[
I_1 = \left[ (x + y)z - \frac{z^2}{2} \right]_0^{\frac{1}{2}y}
\]

Substituting \( z = \frac{1}{2}y \):

\[
I_1 = (x + y) \cdot \frac{1}{2}y - \frac{\left(\frac{1}{2}y\right)^2}{2}
\]

Simplify:

\[
I_1 = \frac{1}{2}y(x + y) - \frac{1}{8}y^2
\]

Thus, the result of the first integration is:

\[
I_1 = \frac{1}{2}y(x + y) - \frac{1}{8}y^2
\]

### Step 2: Evaluate the second integral with respect to \( y \)

\[
I_2 = \int_0^2 \left[ \frac{1}{2}y(x + y) - \frac{1}{8}y^2 \right] dy
\]

Expand the integrand:

\[
I_2 = \int_0^2 \left[ \frac{1}{2}xy + \frac{1}{2}y^2 - \frac{1}{8}y^2 \right] dy
\]

Simplify:

\[
I_2 = \int_0^2 \left[ \frac{1}{2}xy + \frac{3}{8}y^2 \right] dy
\]

Now, integrate term by term:

\[
\int_0^2 \frac{1}{2}xy \, dy = \frac{1}{2}x \cdot \frac{y^2}{2} \Big|_0^2 = \frac{1}{2}x \cdot 2 = x
\]

\[
\int_0^2 \frac{3}{8}y^2 \, dy = \frac{3}{8} \cdot \frac{y^3}{3} \Big|_0^2 = \frac{3}{8} \cdot \frac{8}{3} = 1
\]

So, the result of the second integration is:

\[
I_2 = x + 1
\]

### Step 3: Evaluate the outermost integral with respect to \( x \)

\[
I_3 = \int_0^1 (x + 1) dx
\]

This is straightforward:

\[
\int_0^1 (x + 1) dx = \frac{x^2}{2} + x \Big|_0^1 = \frac{1}{2} + 1 = \frac{3}{2}
\]

### Final Answer

The value of the triple integral is:

\[
\frac{3}{2}
\]

Would you like further clarifications or details on any of the steps? 

Here are some related questions for further exploration:
1. How would the limits of integration affect the result if they were changed?
2. What is the geometric interpretation of this triple integral?
3. How does the integrand \( x + y - z \) influence the complexity of the solution?
4. Could this problem be solved using a different method, like polar or spherical coordinates?
5. What would the answer be if the integrand were just \( x + y \)?

**Tip:** Always start by carefully evaluating the innermost integral and simplify before moving on to the next steps.

Evaluate the triple integral \( \int_0^1 \int_0^2 \int_0^{\frac{1}{2}y} (x + y - z) dz dy dx \).

Learn how to solve a triple integral \( \int_0^1 \int_0^2 \int_0^{\frac{1}{2}y} (x + y - z) dz dy dx \). This step-by-step solution breaks down each integral from the innermost to the outermost layer. Understand key concepts in multivariable calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation