Let The Region \[$ R \$\] Be The Area Enclosed By The Function \[$ F(x) = E^x - 2 \$\], The Horizontal Line \[$ Y = -4 \$\], And The Vertical Lines \[$ X = 0 \$\] And \[$ X = 3 \$\]. Find The Volume Of The Solid

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Let the Region { R $}$ be the Area Enclosed by the Function { f(x) = e^x - 2 $}$, the Horizontal Line { y = -4 $}$, and the Vertical Lines { x = 0 $}$ and { x = 3 $}$. Find the Volume of the Solid

In this article, we will explore the problem of finding the volume of a solid enclosed by a function, a horizontal line, and two vertical lines. The function in question is { f(x) = e^x - 2 $}$, and the horizontal line is { y = -4 $}$. The vertical lines are { x = 0 $}$ and { x = 3 $}$. We will use the method of integration to find the volume of the solid.

To begin, let's understand the problem at hand. We are given a function { f(x) = e^x - 2 $}$ and a horizontal line { y = -4 $}$. We are also given two vertical lines { x = 0 $}$ and { x = 3 $}$. The region { R $}$ is the area enclosed by these curves. Our goal is to find the volume of the solid formed by rotating this region about the x-axis.

To visualize the region, let's first graph the function { f(x) = e^x - 2 $}$ and the horizontal line { y = -4 $}$. We can see that the function is an exponential curve that opens upwards, and the horizontal line is a flat line at { y = -4 $}$. The vertical lines { x = 0 $}$ and { x = 3 $}$ are the boundaries of the region.

To find the volume of the solid, we will use the method of integration. We will integrate the area of the region with respect to the x-axis. The area of the region is given by the function { f(x) = e^x - 2 $}$ minus the horizontal line { y = -4 $}$. So, the area of the region is { A(x) = e^x - 2 - (-4) = e^x + 2 $}$.

The method of integration is a powerful tool for finding the volume of a solid. The basic idea is to integrate the area of the region with respect to the x-axis. In this case, we will integrate the area of the region from { x = 0 $}$ to { x = 3 $}$.

The integral we need to evaluate is { \int_{0}^{3} (e^x + 2) dx $}$. To evaluate this integral, we can use the fundamental theorem of calculus. The fundamental theorem of calculus states that the definite integral of a function is equal to the antiderivative of the function evaluated at the upper limit minus the antiderivative of the function evaluated at the lower limit.

To evaluate the integral, we can use the antiderivative of the function { e^x + 2 $}$. The antiderivative of { e^x $}$ is { e^x $}$, and the antiderivative of { 2 $}$ is { 2x $}$. So, the antiderivative of { e^x + 2 $}$ is { e^x + 2x $}$.

Now, we can apply the fundamental theorem of calculus to evaluate the integral. We have { \int_{0}^{3} (e^x + 2) dx = (e^x + 2x) \big|_{0}^{3} $}$. Evaluating this expression, we get { (e^3 + 6) - (e^0 + 2(0)) = e^3 + 6 - 1 = e^3 + 5 $}$.

The volume of the solid is given by the integral we evaluated. So, the volume of the solid is { V = e^3 + 5 $}$.

In this article, we explored the problem of finding the volume of a solid enclosed by a function, a horizontal line, and two vertical lines. We used the method of integration to find the volume of the solid. The volume of the solid is given by the integral we evaluated, which is { V = e^3 + 5 $}$. This result provides a solution to the problem and demonstrates the power of the method of integration in finding the volume of a solid.

The final answer is: e3+5\boxed{e^3 + 5}
Q&A: Finding the Volume of a Solid Enclosed by a Function, a Horizontal Line, and Two Vertical Lines

In our previous article, we explored the problem of finding the volume of a solid enclosed by a function, a horizontal line, and two vertical lines. We used the method of integration to find the volume of the solid. In this article, we will answer some common questions related to this problem.

A: The region { R $}$ is the area enclosed by the function { f(x) = e^x - 2 $}$, the horizontal line { y = -4 $}$, and the vertical lines { x = 0 $}$ and { x = 3 $}$.

A: We need to find the volume of the solid because it is a fundamental concept in mathematics and physics. The volume of a solid is a measure of its size and is used in a wide range of applications, including engineering, architecture, and science.

A: The method of integration is a powerful tool for finding the volume of a solid. It involves integrating the area of the region with respect to the x-axis.

A: To evaluate the integral, we can use the fundamental theorem of calculus. The fundamental theorem of calculus states that the definite integral of a function is equal to the antiderivative of the function evaluated at the upper limit minus the antiderivative of the function evaluated at the lower limit.

A: The antiderivative of { e^x $}$ is { e^x $}$, and the antiderivative of { 2 $}$ is { 2x $}$. So, the antiderivative of { e^x + 2 $}$ is { e^x + 2x $}$.

A: We can apply the fundamental theorem of calculus by evaluating the antiderivative of the function at the upper limit and subtracting the antiderivative of the function evaluated at the lower limit.

A: The final answer to the problem is { V = e^3 + 5 $}$.

In this article, we answered some common questions related to finding the volume of a solid enclosed by a function, a horizontal line, and two vertical lines. We hope that this article has provided a clear understanding of the problem and its solution.

The final answer is: e3+5\boxed{e^3 + 5}