6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
c. Write an integral for the volume of the solid using the shell method.

검증된 단계별 안내

Step 1: Understand the shell method. The shell method involves integrating the volume of cylindrical shells formed by revolving a region around a vertical or horizontal axis. The formula for the volume is: \( V = \int_{a}^{b} 2\pi \cdot \text{radius} \cdot \text{height} \cdot dx \).

Step 2: Identify the radius and height of the shells. Since the region R is revolved around the line \( x = 4 \), the radius of a shell is \( 4 - x \) (distance from the line \( x = 4 \) to a point \( x \) in the region). The height of the shell is given by the difference between the upper curve \( y = 2 \) and the lower curve \( y = 2 - \sqrt{x} \).

Step 3: Set up the integral bounds. The region R is bounded horizontally between \( x = 0 \) and \( x = 4 \). These will be the limits of integration.

Step 4: Write the integral expression for the volume. Using the shell method formula, the integral becomes: \( V = \int_{0}^{4} 2\pi (4 - x) \cdot [(2) - (2 - \sqrt{x})] \cdot dx \).

Step 5: Simplify the height expression inside the integral. The height \( (2) - (2 - \sqrt{x}) \) simplifies to \( \sqrt{x} \). Thus, the integral for the volume becomes: \( V = \int_{0}^{4} 2\pi (4 - x) \cdot \sqrt{x} \cdot dx \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Region Bounded by Curves

In this problem, the region R is defined by the curves y = 2, y = 2 - √x, and the vertical line x = 4. Understanding how to identify and sketch the area bounded by these curves is crucial for visualizing the solid of revolution. The intersection points of these curves help determine the limits of integration when calculating volume.

Shell Method

The shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into cylindrical shells and integrating the volume of these shells. When revolving around a vertical line, the formula for the volume V is given by V = 2π ∫ (radius)(height) dx, where the radius is the distance from the axis of rotation to the shell and height is the function value.

Setting Up the Integral

To write the integral for the volume using the shell method, one must determine the appropriate limits of integration and the expressions for the radius and height. In this case, the radius is (4 - x) and the height is (2 - (2 - √x)) = √x. The integral will be set up from x = 0 to x = 4, capturing the entire region R as it revolves around the line x = 4.

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.c. Write an integral for the volume of the solid using the shell method.