91Ó°ÊÓ

The Cylindrical Shells Method is a powerful technique used to find the volume of a solid of revolution. This is particularly useful when revolving a region around a vertical line, such as the y-axis. To intuitively understand this method, imagine peeling an onion: each thin layer you remove represents a cylindrical shell. By adding the volumes of these shells, you get the total volume of the solid.

The key formula for finding the volume when revolving around the y-axis is:

• \( V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \)

Here, \( x \) is the radius of each shell, and \( f(x) \) is the height. For our problem, the function \( f(x) = e^x \), and we consider the interval from \( x = 1 \) to \( x = 2 \). This means we are looking at a region under the curve from 1 to 2.

• Think of each slice as a wrapper around the y-axis, stretching from \( x = 1 \) to \( x = 2 \).
• Multiply the results by \( 2\pi \) to account for the circular nature of the revolution.

With this method, each shell's outer curve contributes to the total volume, and by setting up the integral across the specified limits, we can sum them up efficiently.

Integration by Parts is a vital technique in calculus for integrating products of functions. The formula is derived from the product rule for differentiation and provides a way to transform difficult integrals into simpler ones.

The integration by parts formula is:

• \( \int u \, dv = uv - \int v \, du \)

Here’s how we used it for \( \int x \, e^x \, dx \):

• Choose \( u = x \), which simplifies to \( du = dx \).
• Select \( dv = e^x \, dx \), giving \( v = e^x \) after integration.

Plug these into the formula:

• \( \int x \, e^x \, dx = x \, e^x - \int e^x \, dx \)

The second integral is \( \int e^x \, dx = e^x \), a straightforward antiderivative.

Ultimately, the integration simplifies to \( x \, e^x - e^x + C \). Evaluating this from 1 to 2, it becomes \( \left[ x \, e^x - e^x \right]_{1}^{2} \), providing the definite result needed for finding the volume. This method allowed us to effectively handle the complexities arising from products of functions.

Definite Integrals are a fundamental concept used to find the area under a curve over a given interval. In the context of our problem, they help compute the volume of the solid after applying the cylindrical shells method.

The notation for a definite integral is:

• \( \int_{a}^{b} f(x) \, dx \)

Where \( a \) and \( b \) are the limits of integration. This represents the area under the curve \( f(x) \) from \( x = a \) to \( x = b \). In the exercise, we integrated \( x \, e^x \) from \( x = 1 \) to \( x = 2 \).

Once the integral is evaluated, it is crucial to apply the boundaries:

• \( \left[ x \, e^x - e^x \right]_{1}^{2} \)
• Calculate by substituting \( x = 2 \) and \( x = 1 \) into \( x \, e^x - e^x \) and finding the difference.

For this exercise, this calculation resulted in the expression \( e^2 - e \). When multiplied by the leading constants from the cylindrical shells method, specifically \( 2\pi \), it yields the final volume. Definite integrals thus provided the precise calculation required to determine the volume between the specified limits.