Volumes
Find the volumes of the solids in Exercises 1–18.
The solid lies between planes perpendicular to the x-axis at x = π/4 and x = 5π/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.

검증된 단계별 안내

Identify the interval over which the volume is to be found: from \(x = \frac{\pi}{4}\) to \(x = \frac{5\pi}{4}\).

Determine the diameter of each circular cross-section at a given \(x\). The diameter is the vertical distance between the two curves, so calculate \(D(x) = |2 \sin x - 2 \cos x|\).

Express the radius of the circular cross-section as half the diameter: \(r(x) = \frac{D(x)}{2} = \frac{|2 \sin x - 2 \cos x|}{2} = |\sin x - \cos x|\).

Write the area of each circular cross-section as a function of \(x\): \(A(x) = \pi [r(x)]^2 = \pi (\sin x - \cos x)^2\) (since squaring removes the absolute value).

Set up the volume integral by integrating the cross-sectional area along the \(x\)-axis between the given bounds: \(V = \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \pi (\sin x - \cos x)^2 \, dx\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

주요 개념

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

Volume of Solids with Known Cross-Sections

This concept involves finding the volume of a solid by integrating the area of its cross-sections perpendicular to an axis. When cross-sections are circular disks, the area is π times the radius squared. The volume is obtained by integrating these areas over the given interval along the axis.

Determining the Radius from Two Curves

The radius of each circular cross-section is half the distance between the two curves defining the diameter. Here, the diameter is the vertical distance between y = 2 cos x and y = 2 sin x, so the radius is half the absolute difference of these functions at each x.

Definite Integration with Trigonometric Functions

To find the volume, you integrate the area function involving trigonometric expressions over the interval [π/4, 5π/4]. Understanding how to integrate squares of sine and cosine functions, possibly using identities, is essential for evaluating the integral correctly.

VolumesFind the volumes of the solids in Exercises 1–18.The solid lies between planes perpendicular to the x-axis at x = π/4 and x = 5π/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.