Evaluate double integral of x^m-1 y^n-1 dxdy over the region bounded by x+y=h ,x=0& y=0

We are asked to evaluate the double integral of \( x^{m-1} y^{n-1} \) over the triangular region bounded by the lines \( x + y = h \), \( x = 0 \), and \( y = 0 \). The integral is:

\[
I = \int_0^h \int_0^{h - y} x^{m-1} y^{n-1} \, dx \, dy
\]

### Step-by-step solution:

1. **Evaluate the inner integral** with respect to \( x \):
   \[
   \int_0^{h-y} x^{m-1} \, dx = \frac{(h-y)^m}{m}
   \]
   (This is the result of integrating a power of \( x \), specifically \( \int x^{m-1} \, dx = \frac{x^m}{m} \)).

2. **Substitute this result into the outer integral**:
   \[
   I = \int_0^h \frac{(h-y)^m}{m} y^{n-1} \, dy
   \]

3. **Now, evaluate the outer integral** with respect to \( y \):
   Let’s perform a substitution to make the integration easier. Set \( u = h - y \), so that \( du = -dy \). When \( y = 0 \), \( u = h \), and when \( y = h \), \( u = 0 \). Thus, the integral becomes:
   \[
   I = \frac{1}{m} \int_h^0 u^m (h - u)^{n-1} (-du)
   \]
   Simplifying:
   \[
   I = \frac{1}{m} \int_0^h u^{m} (h - u)^{n-1} \, du
   \]

4. **Recognize this as a Beta function**:
   The expression:
   \[
   \int_0^h u^{m} (h - u)^{n-1} \, du
   \]
   is of the form of a Beta function, specifically:
   \[
   \int_0^1 t^{m} (1 - t)^{n-1} \, dt = \frac{\Gamma(m+1) \Gamma(n)}{\Gamma(m+n+1)}
   \]
   So, rescaling the variable \( u \) by setting \( t = \frac{u}{h} \), we get:
   \[
   I = \frac{h^{m+n}}{m} \cdot \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}
   \]

### Final Answer:
\[
I = \frac{h^{m+n}}{m \cdot n}
\]

This is the evaluated double integral.

Do you have any questions or would you like further details on any step?

Here are five related questions to extend the concept:
1. How does the Beta function relate to the Gamma function?
2. What is the geometric interpretation of a double integral?
3. How do you transform the limits of integration when changing variables?
4. Can this integral be solved using polar coordinates?
5. How does the integral change if the region of integration is a different triangle?

**Tip:** Whenever dealing with integrals involving power functions, the Beta and Gamma functions are powerful tools to simplify the process.

Evaluate double integral of x^(m-1) y^(n-1) dxdy over the region bounded by x+y=h, x=0, and y=0.

Learn how to evaluate the double integral of x^(m-1) y^(n-1) over the triangular region bounded by x + y = h, x = 0, and y = 0. This detailed solution covers key calculus concepts such as double integration, the Beta and Gamma functions, and variable substitution. Suitable for undergraduate students in Calculus II or III.

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