Thank you for sharing the image. From the screenshot, I see two problems requiring the calculation of the average value of a function over a given rectangle using double integrals. Let's tackle one problem at a time.

Problem 15:

The function and domain are:

• $f(x, y) = 5e^y \sqrt{x + e^y}$
• The rectangle $R = [0, 6] \times [0, 1]$.

The average value of $f$ over $R$ is given by: $f_{\text{avg}} = \frac{1}{\text{Area of } R} \int \int_R f(x, y) \, dA$ Here, the area of $R$ is: $\text{Area of } R = (6 - 0)(1 - 0) = 6$

Substitute into the formula: $f_{\text{avg}} = \frac{1}{6} \int_0^1 \int_0^6 5e^y \sqrt{x + e^y} \, dx \, dy$

Step 1: Integrate with respect to $x$

Inside the integral, treat $y$ as constant. The inner integral becomes: $\int_0^6 5e^y \sqrt{x + e^y} \, dx$ Using substitution, let $u = x + e^y$, so $du = dx$ and the limits become:

• When $x = 0$: $u = e^y$,
• When $x = 6$: $u = 6 + e^y$.

The integral becomes: $\int_0^6 5e^y \sqrt{x + e^y} \, dx = \int_{e^y}^{6 + e^y} 5e^y \sqrt{u} \, du$

Now calculate: $\int 5e^y \sqrt{u} \, du = 5e^y \cdot \frac{2}{3} u^{3/2} = \frac{10e^y}{3} u^{3/2}$

Evaluate from $u = e^y$ to $u = 6 + e^y$: $\frac{10e^y}{3} \left[ (6 + e^y)^{3/2} - (e^y)^{3/2} \right]$

Step 2: Integrate with respect to $y$

Substitute this result into the outer integral: $f_{\text{avg}} = \frac{1}{6} \int_0^1 \frac{10e^y}{3} \left[ (6 + e^y)^{3/2} - (e^y)^{3/2} \right] \, dy$ Simplify: $f_{\text{avg}} = \frac{5}{9} \int_0^1 e^y \left[ (6 + e^y)^{3/2} - (e^y)^{3/2} \right] \, dy$

This integral will require numerical methods or further substitution for evaluation.

Problem 16:

The function and domain are:

• $f(x, y) = 3e^y \sqrt{x + e^y}$
• The rectangle $R = [0, 10] \times [0, 1]$.

Following the same method as above:

1. The area of $R$ is $(10 - 0)(1 - 0) = 10$.
2. The average value formula becomes: $f_{\text{avg}} = \frac{1}{10} \int_0^1 \int_0^{10} 3e^y \sqrt{x + e^y} \, dx \, dy$

The inner integral is: $\int_0^{10} 3e^y \sqrt{x + e^y} \, dx$ Using the same substitution $u = x + e^y$, calculate and proceed similarly to Problem 15.

Would you like me to simplify the computations further for any of these problems? Let me know!

Related Questions:

1. How do you set up double integrals for average value calculations?
2. What is the substitution method in integrals, and how does it simplify problems like these?
3. Can numerical integration techniques help solve complex integrals more efficiently?
4. What is the importance of understanding the bounds of integration in these problems?
5. How can software tools like MATLAB or Python be used to evaluate such integrals?

Tip: Always break down double integrals into simpler single integrals by carefully choosing the order of integration!