Region R is revolved about the line x=4 to form a solid of revolution.

a. What is the radius of a cross section of the solid at a point y in [1, 3]?

검증된 단계별 안내

Step 1: Understand the problem setup. The region R is revolved about the vertical line x=4, forming a solid of revolution. The radius of a cross-section at a given y-value is the horizontal distance between the line x=4 and the x-coordinate of the region R at that y-value.

Step 2: Identify the x-coordinate of the region R at a given y-value. If the region R is defined by a function x=f(y), then the x-coordinate at a specific y-value is given by f(y).

Step 3: Calculate the radius of the cross-section. The radius is the absolute difference between the line x=4 and the x-coordinate of the region R. Mathematically, this is expressed as:

| 4 - f (y) |

Step 4: Ensure the y-value is within the given interval [1, 3]. The radius calculation is valid only for y-values in this range, as specified in the problem.

Step 5: Substitute the specific function f(y) (if provided) into the formula for the radius to express the radius explicitly in terms of y. If no function is given, the radius remains in the general form:

| 4 - f (y) |

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주요 개념

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

Solid of Revolution

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional area around a straight line (axis of rotation). The shape's volume can be calculated using methods such as the disk or washer method, depending on the orientation of the axis and the region being revolved.

Radius of Cross Section

The radius of a cross section of a solid of revolution is the distance from the axis of rotation to the edge of the region being revolved. This radius varies depending on the specific point along the axis and is crucial for calculating the volume of the solid using integration.

Integration in Volume Calculation

Integration is a fundamental concept in calculus used to find the volume of solids of revolution. By integrating the area of cross sections along the axis of rotation, one can determine the total volume of the solid. The specific method of integration (disk, washer, or cylindrical shells) depends on the geometry of the solid and the axis of rotation.

Region R is revolved about the line x=4 to form a solid of revolution. a. What is the radius of a cross section of the solid at a point y in [1, 3]?