Question

Find the volume of the solid generated by revolving the region in the first quadrant bounded by \(y=e^{x}\) and the \(x\) -axis, from \(x=0\) to \(x=\ln (7)\), about the \(y\) -axis. (Express the answer in exact form.)

Short answer

The volume is \( 12\pi \).

Step-by-step solution

Set up the problem context

We need to find the volume of a solid generated by revolving a region about the y-axis. The region is bounded by the curve \( y = e^x \), the x-axis, and vertical lines \( x = 0 \) and \( x = \ln(7) \).

Choose the integration method

Since we are revolving around the y-axis, it is convenient to use the shell method. The volume \( V \) of the solid of revolution is given by the formula: \[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \] where \( f(x) \) is the height of the shell and \( x \) is the radius.

Set up the integral limits and function

Our region is bounded by \( x = 0 \) and \( x = \ln(7) \), so the limits of integration are from 0 to \( \ln(7) \). The function defining the height of the shell is \( f(x) = e^x \).

Formulate the integral

Substitute the function and limits into the shell method formula: \[ V = 2\pi \int_{0}^{\ln(7)} x \cdot e^x \, dx \]

Integrate the function

To integrate \( x \cdot e^x \), use integration by parts where \( u = x \) and \( dv = e^x \, dx \). Then \( du = dx \) and \( v = e^x \). The integration by parts formula is \( \int u \, dv = uv - \int v \, du \).

Compute the integration by parts

Applying integration by parts: \[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C \].

Evaluate the definite integral

Evaluate from 0 to \( \ln(7) \): \[ \left[ (x e^x - e^x) \right]_0^{\ln(7)} = \left[ (\ln(7) e^{\ln(7)} - e^{\ln(7)}) - (0 - 1) \right] = 6. \]

Calculate the volume

Now, substitute the integrated result back into the volume formula: \[ V = 2\pi \times 6 = 12\pi. \]

Finalize the solution

Our volume is therefore \( 12\pi \).

Key concepts

Shell Method

The shell method is a technique used to find the volume of a solid of revolution, particularly useful when revolving a region around a vertical axis, like the y-axis. Unlike the disk method which uses horizontal slices, the shell method involves the concept of cylindrical "shells." The idea is to integrate around the height and radius of these shells.To apply the shell method:

• Identify the radius of the shell, which is the distance from the axis of rotation to the shell. In this exercise, it is represented by the variable \( x \).
• Determine the height of the shell, given by the function \( f(x) \). Here, it is \( f(x) = e^x \) because it outlines the height of our region.
• Use the shell method formula \( V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \), where \( a \) and \( b \) are the bounds of the region.

This method elegantly simplifies problems when working with transformations around the y-axis, as demonstrated in the original solution.

Integration by Parts

Integration by parts is a technique in integral calculus used when integrating products of functions, especially when they cannot be directly integrated. This is particularly necessary when dealing with integrals like \( x \cdot e^x \), as seen in our exercise.The integration by parts formula is:\[ \int u \, dv = uv - \int v \, du \]Here's how it works with our example:

• Choose \( u = x \); hence \( du = dx \).
• Let \( dv = e^x \, dx \); consequently, \( v = e^x \).

Applying this combination to the formula, you integrate by parts:\[ \int x e^x \, dx = x e^x - \int e^x \, dx \]This simplifies the integration process by subsequently tackling the simpler integral \( \int e^x \, dx = e^x \). This approach is vital in problems where simple integration is not feasible.

Definite Integral

A definite integral calculates the area under a curve between two specified limits, or more generally, the net "accumulation" of a quantity. Unlike indefinite integrals, definite integrals result in a specific numerical value.In our problem, computing the definite integral:\[ \int_{0}^{\ln(7)} x e^x \, dx \]This integral represents the sum of little slices under the curve \( y = x \cdot e^x \) from \( x = 0 \) to \( x = \ln(7) \). This operation yields the net volume as a concrete value, not just an expression, which is crucial for determining the volume of revolution.

Exponential Function

The exponential function \( e^x \) is one of the most significant functions in mathematics, often appearing in different areas such as growth models, natural phenomena, and calculus problems as seen here.Key features of the exponential function include:

• Its base, Euler's number \( e \), approximately 2.718, which is an irrational constant.
• It grows rapidly, meaning as \( x \) increases, \( e^x \) increases exponentially.
• Its derivative is \( e^x \), which makes integration and differentiation straightforward.

In the volume of revolution problem, \( e^x \) defines both the boundary and height of the region being revolved, fundamentally shaping the volume we calculate.