13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.

ρ(x)=1+sin x, for 0≤x≤π

검증된 단계별 안내

Identify the given density function \( \rho(x) = 1 + \sin x \) and the interval over which the bar extends, which is \( 0 \leq x \leq \pi \).

Recall that the mass \( M \) of a one-dimensional object with density function \( \rho(x) \) over the interval \( [a, b] \) is given by the integral \( M = \int_a^b \rho(x) \, dx \).

Set up the integral for the mass using the given density and interval: \( M = \int_0^{\pi} (1 + \sin x) \, dx \).

Break the integral into two simpler integrals: \( M = \int_0^{\pi} 1 \, dx + \int_0^{\pi} \sin x \, dx \).

Evaluate each integral separately: the integral of 1 over \( [0, \pi] \) and the integral of \( \sin x \) over \( [0, \pi] \), then sum the results to find the total mass.

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주요 개념

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

Density Function

The density function ρ(x) describes how mass is distributed along the length of the bar. It gives the mass per unit length at each point x, allowing us to calculate total mass by integrating over the interval.

Definite Integral for Mass

The total mass of a one-dimensional object with variable density is found by integrating the density function over the given interval. Specifically, mass = ∫ from a to b of ρ(x) dx, summing all infinitesimal mass elements.

Integration of Trigonometric Functions

Since the density function includes sin x, understanding how to integrate trigonometric functions is essential. The integral of sin x over an interval can be computed using standard antiderivatives, facilitating the calculation of total mass.

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.ρ(x)=1+sin x, for 0≤x≤π