Use the general slicing method to find the volume of the following solids.
The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares

검증된 단계별 안내

Step 1: Understand the problem. The base of the solid is the region bounded by the semicircle y = √(1 − x²) and the x-axis. The cross-sections perpendicular to the x-axis are squares. This means the side length of each square is determined by the height of the semicircle at a given x-coordinate.

Step 2: Express the side length of the square. The side length of the square at a given x-coordinate is twice the y-value of the semicircle, which is 2√(1 − x²).

Step 3: Write the formula for the area of the square. The area of a square is the square of its side length. Therefore, the area of the square at a given x-coordinate is [2√(1 − x²)]² = 4(1 − x²).

Step 4: Set up the integral for the volume. The volume of the solid is obtained by integrating the area of the cross-sections along the x-axis over the interval where the semicircle exists. The semicircle spans from x = −1 to x = 1. Thus, the volume is given by V = ∫[−1, 1] 4(1 − x²) dx.

Step 5: Simplify the integral. Break the integral into simpler parts: V = ∫[−1, 1] 4 dx − ∫[−1, 1] 4x² dx. Evaluate each term separately to find the volume.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Volume of Solids of Revolution

The volume of solids of revolution can be calculated using integration techniques, particularly the disk or washer method. In this case, the solid is formed by rotating a region around an axis, and the volume is determined by integrating the area of cross-sectional slices perpendicular to that axis. Understanding this concept is crucial for applying the slicing method effectively.

Cross Sections

Cross sections are the shapes obtained by slicing a three-dimensional object with a plane. In this problem, the cross sections of the solid are squares, which means that the area of each cross section can be expressed as a function of the variable x. This relationship is essential for setting up the integral to find the volume of the solid.

Integration

Integration is a fundamental concept in calculus used to find areas, volumes, and other quantities that accumulate over a range. In this context, integration will be used to sum the areas of the square cross sections along the x-axis, from the leftmost to the rightmost bounds of the semicircle. Mastery of integration techniques is necessary to compute the total volume of the solid.

Use the general slicing method to find the volume of the following solids.The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares