Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures).

b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2].

검증된 단계별 안내

Identify the region R bounded by the curve \(y = \cos^{-1} x\) and the x-axis on the interval \([0,1]\). Since \(y = \cos^{-1} x\), we can rewrite this as \(x = \cos y\) for \(y\) in \([0, \frac{\pi}{2}]\) because \(\cos^{-1} x\) maps \([0,1]\) to \([0, \frac{\pi}{2}]\).

Understand that the solid is formed by revolving the region R about the y-axis. For a fixed \(y\) in \([0, \frac{\pi}{2}]\), the cross section perpendicular to the y-axis is a circle.

Determine the radius of the cross-sectional circle at height \(y\). Since the solid is revolved around the y-axis, the radius is the horizontal distance from the y-axis to the curve, which is \(x = \cos y\).

Write the formula for the area \(A(y)\) of the cross section at height \(y\). The area of a circle is \(\pi\) times the radius squared, so \(A(y) = \pi (\cos y)^2\).

Express the final formula for the cross-sectional area as \(A(y) = \pi \cos^2 y\) for \(y\) in \([0, \frac{\pi}{2}]\).

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.

Inverse Trigonometric Functions

The function y = cos⁻¹(x) is the inverse cosine function, which maps values from [0,1] to angles in [0, π/2]. Understanding its domain and range is essential to relate x and y coordinates and to express x as a function of y when analyzing the region and cross sections.

Cross-Sectional Area of a Solid of Revolution

A cross section perpendicular to the axis of revolution is a shape whose area depends on the distance from the axis. For revolution about the y-axis, the cross section at height y is typically a disk or washer, and its area A(y) can be found using the radius determined by the x-values corresponding to y.

Relationship Between x and y in the Region

Since y = cos⁻¹(x), we can rewrite x in terms of y as x = cos(y). This relationship allows us to express the radius of the cross section at height y, which is crucial for setting up the area formula A(y) for the solid's cross section.

Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures). b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2].