Use Double Integrals To Compute The Area Of The Region In The First Quadrant Bounded By Y = E X Y=e^x Y = E X And X = Ln ⁡ 14 X=\ln 14 X = Ln 14 .The Area Of The Region Is □ \square □ . (Simplify Your Answer.)

Introduction

In mathematics, double integrals are a powerful tool for calculating the area of complex regions. In this article, we will use double integrals to compute the area of the region in the first quadrant bounded by the exponential function $y=e^x$ and the logarithmic function $x=\ln 14$. This problem requires a deep understanding of integration and the ability to apply it to real-world scenarios.

Understanding the Problem

The problem asks us to find the area of the region bounded by the exponential function $y=e^x$ and the logarithmic function $x=\ln 14$. To solve this problem, we need to understand the behavior of these two functions and how they intersect in the first quadrant.

The exponential function $y=e^x$ is a continuous and increasing function that passes through the point $(0,1)$. It has a horizontal asymptote at $y=0$ and a vertical asymptote at $x=-\infty$. The logarithmic function $x=\ln 14$ is a continuous and increasing function that passes through the point $(\ln 14,0)$. It has a horizontal asymptote at $x=-\infty$ and a vertical asymptote at $y=0$.

Setting Up the Double Integral

To compute the area of the region bounded by these two functions, we need to set up a double integral. The double integral will be of the form:

$$\int_{a}^{b} \int_{f(x)}^{g(x)} 1 \,dy\,dx$$

where $a$ and $b$ are the limits of integration for the $x$ variable, and $f(x)$ and $g(x)$ are the functions that bound the region.

In this case, the limits of integration for the $x$ variable are $a=0$ and $b=\ln 14$. The functions that bound the region are $f(x)=e^x$ and $g(x)=\ln 14$.

Evaluating the Double Integral

To evaluate the double integral, we need to integrate the function $1$ with respect to $y$ and then integrate the result with respect to $x$.

The inner integral is:

$$\int_{f(x)}^{g(x)} 1 \,dy = \int_{e^x}^{\ln 14} 1 \,dy = \ln 14 - e^x$$

The outer integral is:

$$\int_{a}^{b} (\ln 14 - e^x) \,dx = \int_{0}^{\ln 14} (\ln 14 - e^x) \,dx$$

Solving the Outer Integral

To solve the outer integral, we need to integrate the function $\ln 14 - e^x$ with respect to $x$.

Using the properties of integration, we can write:

$$\int_{0}^{\ln 14} (\ln 14 - e^x) \,dx = \ln 14 \int_{0}^{\ln 14} 1 \,dx - \int_{0}^{\ln 14} e^x \,dx$$

Evaluating the First Integral

The first integral is:

$$\int_{0}^{\ln 14} 1 \,dx = x \Big|_{0}^{\ln 14} = \ln 14 - 0 = \ln 14$$

Evaluating the Second Integral

The second integral is:

$$\int_{0}^{\ln 14} e^x \,dx = e^x \Big|_{0}^{\ln 14} = e^{\ln 14} - e^0 = 14 - 1 = 13$$

Combining the Results

Now we can combine the results of the two integrals:

$$\int_{0}^{\ln 14} (\ln 14 - e^x) \,dx = \ln 14 \int_{0}^{\ln 14} 1 \,dx - \int_{0}^{\ln 14} e^x \,dx = \ln 14 (\ln 14) - 13 = 14\ln 14 - 13$$

Conclusion

In this article, we used double integrals to compute the area of the region in the first quadrant bounded by the exponential function $y=e^x$ and the logarithmic function $x=\ln 14$. The area of the region is given by the expression $14\ln 14 - 13$. This problem required a deep understanding of integration and the ability to apply it to real-world scenarios.

Final Answer

The final answer is: $\boxed{14\ln 14 - 13}$

Q: What is the main concept behind computing the area of a region bounded by exponential and logarithmic functions?

A: The main concept behind computing the area of a region bounded by exponential and logarithmic functions is the use of double integrals. Double integrals are a powerful tool for calculating the area of complex regions.

Q: What are the limits of integration for the x variable in this problem?

A: The limits of integration for the x variable are a=0 and b=ln 14.

Q: What are the functions that bound the region in this problem?

A: The functions that bound the region are f(x)=e^x and g(x)=ln 14.

Q: How do you evaluate the double integral in this problem?

A: To evaluate the double integral, we need to integrate the function 1 with respect to y and then integrate the result with respect to x.

Q: What is the inner integral in this problem?

A: The inner integral is ∫[f(x)]^g(x) 1 dy = ∫[e^x]ln 14 1 dy = ln 14 - e^x.

Q: What is the outer integral in this problem?

A: The outer integral is ∫[a]^b (ln 14 - e^x) dx = ∫[0]^ln 14 (ln 14 - e^x) dx.

Q: How do you solve the outer integral in this problem?

A: To solve the outer integral, we need to integrate the function ln 14 - e^x with respect to x.

Q: What is the final answer to this problem?

A: The final answer is 14ln 14 - 13.

Q: What is the significance of this problem?

A: This problem is significant because it demonstrates the use of double integrals to compute the area of a complex region bounded by exponential and logarithmic functions.

Q: What are some real-world applications of this problem?

A: Some real-world applications of this problem include calculating the area of a region bounded by two curves, calculating the volume of a solid, and calculating the surface area of a solid.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

• Not setting up the double integral correctly
• Not evaluating the inner integral correctly
• Not solving the outer integral correctly
• Not simplifying the final answer correctly

Q: How can I practice solving this type of problem?

A: You can practice solving this type of problem by:

• Working through example problems
• Practicing with different types of functions
• Using online resources and calculators to check your work
• Seeking help from a teacher or tutor if you are struggling.