Questions: Find the mass of the thin bar with the given density function. ρ(x)=1+sin x ; for 0 ≤ x ≤ π/3 Set up the integral that gives the mass of the thin bar.

Find the mass of the thin bar with the given density function. ρ(x)=1+sin x ; for 0 ≤ x ≤ π/3 Set up the integral that gives the mass of the thin bar.
Transcript text: Find the mass of the thin bar with the given density function. \[ \rho(x)=1+\sin x ; \text { for } 0 \leq x \leq \frac{\pi}{3} \] Set up the integral that gives the mass of the thin bar.
failed

Solution

failed
failed

Solution Steps

To find the mass of the thin bar with the given density function, we need to integrate the density function over the given interval. The mass \( M \) of the bar can be found by evaluating the integral of the density function \(\rho(x)\) from \(0\) to \(\frac{\pi}{3}\).

Step 1: Define the Density Function and Limits

The density function of the thin bar is given by \(\rho(x) = 1 + \sin x\). We need to find the mass of the bar over the interval \(0 \leq x \leq \frac{\pi}{3}\).

Step 2: Set Up the Integral

To find the mass \(M\) of the bar, we set up the integral of the density function over the given interval: \[ M = \int_{0}^{\frac{\pi}{3}} (1 + \sin x) \, dx \]

Step 3: Evaluate the Integral

Evaluate the integral: \[ M = \left[ x - \cos x \right]_{0}^{\frac{\pi}{3}} \]

Step 4: Calculate the Definite Integral

Substitute the limits of integration: \[ M = \left( \frac{\pi}{3} - \cos\left(\frac{\pi}{3}\right) \right) - \left( 0 - \cos(0) \right) \] \[ M = \left( \frac{\pi}{3} - \frac{1}{2} \right) - (-1) \] \[ M = \frac{\pi}{3} - \frac{1}{2} + 1 \] \[ M = \frac{\pi}{3} + \frac{1}{2} \]

Final Answer

\(\boxed{M = \frac{\pi}{3} + \frac{1}{2}}\)

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
f(x)=x^2-5x, g(x)=3x+1, and h(x)=(x-1)/x a) (g ∘ f)(x) b) (h ∘ g)(x) c) (f ∘ g)(x)
failedAnswered
The nurse finds that a visually impaired client is having difficulty in determining which medications to take after being discharged from the hospital. Which intervention would be best in this situation? Filling and labeling the medication bottles for the client Advising the caregiver to administer the client's medication Recommending that the client's pharmacy relabel the medication in large letters Showing the client examples of pill organizers that will help the client sort the medication
failedAnswered
Put the steps in order to solve a quadratic inequality: First Get the inequality in standard form Factor Identify the critical points Express the solution Test the intervals Determine the solution Last
failedAnswered
After operating for several months, architect Barbara Brown completed the following transactions during the latter part of October: View the transactions. Journalize the transactions of Barbara Brown, Architect. Include an explanation with each journal entry. Borrowed money from the bank On October 22, performed services for clients on account totaling 17,600
failedAnswered
Noah draws triangle ABC. Then, he translates that triangle using the rule (x+2, y-1) to create triangle A'B'C'. The measure of angle A is 60 degrees. What is the measure of angle A' in degrees?