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.

b. What is the height of a cylindrical shell at a point x in [0, 4]?

검증된 단계별 안내

Step 1: Understand the problem. The region R is bounded by the curves y = 2−√x, y = 2, and x = 4 in the first quadrant. We are tasked with finding the height of a cylindrical shell at a point x in [0, 4] when the region is revolved about the line x = 4 using the shell method.

Step 2: Recall the shell method formula. The height of a cylindrical shell is determined by the vertical distance between the top curve and the bottom curve at a given x-coordinate. In this case, the top curve is y = 2 and the bottom curve is y = 2−√x.

Step 3: Calculate the height of the shell. The height is given by the difference between the top curve and the bottom curve: height = y_top − y_bottom. Substituting the equations, height = 2 − (2−√x).

Step 4: Simplify the expression for the height. Perform the subtraction: height = 2 − 2 + √x, which simplifies to height = √x.

Step 5: Conclude that the height of the cylindrical shell at a point x in [0, 4] is √x. This height will be used in the shell method formula to compute the volume of the solid.

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주요 개념

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

Cylindrical Shell Method

The cylindrical shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into thin cylindrical shells, which are then integrated to find the total volume. When revolving around a vertical line, the height of each shell is determined by the function defining the region, and the radius is the distance from the axis of rotation.

Height of a Shell

In the context of the shell method, the height of a cylindrical shell at a point x is given by the difference between the upper and lower bounding functions of the region being revolved. For the given region R, the height is calculated as the vertical distance between the line y = 2 and the curve y = 2 - √x, specifically expressed as h(x) = 2 - (2 - √x) = √x.

Bounded Region

A bounded region in calculus refers to a specific area enclosed by curves or lines on a graph. In this case, region R is bounded by the curves y = 2, y = 2 - √x, and the vertical line x = 4. Understanding the boundaries is crucial for accurately applying integration techniques to find volumes or areas related to the region.

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.b. What is the height of a cylindrical shell at a point x in [0, 4]?