Find the average value of f(x,y)=ey4x+ey−−−−−−√
 over the rectangle R=[0,3]×[0,2]
 .


To find the average value of \( f(x, y) = \sqrt{e^{y} 4x + e^y} \) over the rectangle \( R = [0, 3] \times [0, 2] \), we use the formula for the average value of a function over a region \( R \) in two dimensions:

\[
\text{Average value of } f(x, y) = \frac{1}{\text{Area of } R} \iint_R f(x, y) \, dA
\]

### Step 1: Calculate the Area of \( R \)

The area of \( R \) is:
\[
\text{Area of } R = (3 - 0)(2 - 0) = 6
\]

### Step 2: Set up the Integral

We want to compute:
\[
\frac{1}{6} \int_0^3 \int_0^2 \sqrt{e^{y} 4x + e^y} \, dy \, dx
\]

### Step 3: Evaluate the Integral

Rewrite the function inside the integral:
\[
\sqrt{e^{y} 4x + e^y} = \sqrt{e^y (4x + 1)} = e^{y/2} \sqrt{4x + 1}
\]

Thus, the integral becomes:
\[
\frac{1}{6} \int_0^3 \int_0^2 e^{y/2} \sqrt{4x + 1} \, dy \, dx
\]

### Step 4: Integrate with respect to \( y \)

Since \( e^{y/2} \) depends only on \( y \) and \( \sqrt{4x + 1} \) depends only on \( x \), we can separate the integration:
\[
\frac{1}{6} \int_0^3 \sqrt{4x + 1} \left( \int_0^2 e^{y/2} \, dy \right) dx
\]

#### Inner Integral:
\[
\int_0^2 e^{y/2} \, dy = 2 \left( e^{y/2} \right) \Big|_0^2 = 2 \left( e - 1 \right)
\]

So we get:
\[
\frac{1}{6} \int_0^3 \sqrt{4x + 1} \cdot 2(e - 1) \, dx = \frac{e - 1}{3} \int_0^3 \sqrt{4x + 1} \, dx
\]

### Step 5: Integrate with respect to \( x \)

Let \( u = 4x + 1 \), then \( du = 4 \, dx \) or \( dx = \frac{du}{4} \). When \( x = 0 \), \( u = 1 \), and when \( x = 3 \), \( u = 13 \).

The integral becomes:
\[
\frac{e - 1}{3} \int_1^{13} \sqrt{u} \, \frac{du}{4} = \frac{e - 1}{12} \int_1^{13} u^{1/2} \, du
\]

#### Evaluate:
\[
\frac{e - 1}{12} \cdot \frac{2}{3} u^{3/2} \Big|_1^{13} = \frac{e - 1}{18} \left( 13^{3/2} - 1^{3/2} \right)
\]

So, the average value is:
\[
\frac{e - 1}{18} \left( 13^{3/2} - 1 \right)
\]

Would you like more details or have any questions?

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### Related Questions
1. How do we find the average value over regions other than rectangles?
2. How does the presence of \( e^y \) in the function affect the integration?
3. Can the average value formula apply to higher dimensions?
4. How would the answer change if \( R \) had different limits?
5. Why did we separate the integral in Step 4?

### Tip
For average values, simplify the integrand wherever possible before integrating.

Find the average value of f(x,y)=√(e^y 4x + e^y) over the rectangle R=[0,3]×[0,2].

Discover how to find the average value of the function f(x, y) = √(e^y 4x + e^y) over the specified rectangle [0, 3] × [0, 2]. This detailed solution involves multivariable calculus concepts, integral calculus techniques, and key theorems. Ideal for undergraduate students.

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