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Chapter 7: Problem 5

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=\sin (x) \quad I=[\pi / 3, \pi] $$

Short Answer

Expert verified
The average value is \( \frac{9}{4\pi} \).

Step by step solution

01

Identify Formula for Average Value

The formula to find the average value of a function \( f(x) \) over the interval \([a, b]\) is given by: \[ \text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx \]
02

Determine Parameters for the Given Interval

In this problem, the function is \( f(x) = \sin(x) \) and the interval \( I = \left[\frac{\pi}{3}, \pi\right] \). Here, \( a = \frac{\pi}{3} \) and \( b = \pi \).
03

Setup the Integral

Substitute the function and the interval into the formula: \[ \text{Average value} = \frac{1}{\pi - \frac{\pi}{3}} \int_{\frac{\pi}{3}}^{\pi} \sin(x) \, dx \] This simplifies the interval length to \( \frac{2\pi}{3} \).
04

Integrate the Function

Evaluate the integral \( \int_{\frac{\pi}{3}}^{\pi} \sin(x) \, dx \). The antiderivative of \( \sin(x) \) is \( -\cos(x) \): \[ \int \sin(x) \, dx = -\cos(x) + C \] So, \[ \int_{\frac{\pi}{3}}^{\pi} \sin(x) \, dx = \left. -\cos(x) \right|_{\frac{\pi}{3}}^{\pi} = (-\cos(\pi)) - (-\cos(\frac{\pi}{3})) \]
05

Calculate Definite Integral

Substitute the upper and lower limits into the antiderivative: \[ -\cos(\pi) = 1 \] \[ -\cos(\frac{\pi}{3}) = -\left(\frac{1}{2}\right) = -\frac{1}{2} \] Thus, \[ 1 - \left(-\frac{1}{2}\right) = 1 + \frac{1}{2} = \frac{3}{2} \]
06

Calculate the Average Value

Substitute the integral result into the average value formula: \[ \text{Average value} = \frac{1}{\frac{2\pi}{3}} \cdot \frac{3}{2} = \frac{3}{2} \times \frac{3}{2\pi} = \frac{9}{4\pi} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integral
A definite integral is a powerful tool in calculus. It is used to find the accumulated total of a function over a specific interval. The notation \[ \int_a^b f(x) \, dx \]indicates that we are calculating the integral of a function \(f(x)\) from the lower limit \(a\) to the upper limit \(b\). By doing this, we can find the net area between the function curve and the x-axis within this interval.

Here are some key points about definite integrals:
  • The definite integral gives a precise total value as opposed to an antiderivative, which represents a family of functions.
  • To calculate it, you need to find the antiderivative of the function, then apply the fundamental theorem of calculus by plugging in the upper and lower limits.
  • After evaluating their results, you subtract the lower limit evaluation from the upper one to get the final integral value.
This process helps to determine the area under the curve and is crucial in applications such as physics, engineering, and probability.
Trigonometric Functions
Trigonometric functions form a core group of functions in calculus and have applications in various fields including physics and engineering. The sine function \(\sin(x)\) is one such function, characterized by its periodic, wave-like shape.

Here are some important attributes of trigonometric functions, particularly \(\sin(x)\):
  • It varies between -1 and 1, making it a bounded function.
  • The function is periodic with a period of \(2\pi\), meaning \(\sin(x) = \sin(x + 2\pi)\) for any \(x\).
  • Its derivative is \(\cos(x)\) and its antiderivative is \(-\cos(x)\)\, which comes in handy when dealing with integrals such as the one in our exercise.
Understanding these properties not only helps in solving calculus problems but also aids in interpreting phenomena like sound waves or electromagnetic waves where trigonometric functions are used to model oscillations.
Calculus Problem-Solving
Solving problems in calculus involves a step-by-step approach, which includes understanding the problem, setting up equations, and carrying out computations.

For instance, solving a calculus problem to find the average value of a trigonometric function over a given interval involves several steps:
  • Understand the formula: For average value, use \[ \frac{1}{b-a} \int_a^b f(x) \, dx \].
  • Set the parameters: Identify your function \(f(x)\) and the correct limits \([a, b]\).
  • Compute the integral: Find the antiderivative of \(f(x)\) and use the bounds of the interval to evaluate the integral.
  • Perform arithmetic: Use the evaluated integral to calculate the average, ensuring each mathematical operation is correct.
This systematic approach ensures that you not only reach the correct answer but also understand the reasoning behind each step, enhancing comprehension and retention of the Topic.

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Most popular questions from this chapter

A casserole dish at temperature \(T_{0}=180^{\circ} \mathrm{C}\) is plunged into a large basin of hot water whose temperature is \(T_{\infty}=48^{\circ} \mathrm{C}\). The temperature of the dish satisfies Newton's Law of Cooling. In 30 seconds, the temperature of the dish is \(80^{\circ} \mathrm{C}\). How long after the immersion is it before the temperature of the dish cools to \(60^{\circ} \mathrm{C} ?\)

In each of Exercises \(17-28,\) solve the given initial value problem. $$ \frac{d y}{d x}+2=y \quad y(0)=-1 $$

A patient with a fever has temperature at time \(t\) (measured in hours) given by $$ T(t)=99.6-t+0.8 t^{2} \text { degrees } $$ where temperature is measured in degrees Fahrenheit. Find the patient's average temperature as \(t\) ranges from 0 to \(3 .\)

In each of Exercises \(89-92,\) a function \(f\) and an interval \(I\) are given. Calculate the average \(f_{\text {avg }}\) of \(f\) over \(I,\) and find a value \(c\) in \(I\) such that \(f(c)=f_{\text {avg. }}\) State your answers to three decimal places. $$ f(x)=\frac{\ln (x)}{1+x} \quad I=[1, e] $$

Suppose that during wartime, a particular type of weapon is produced at a constant rate \(\mu\) and destroyed in battle at a constant relative rate \(\delta .\) If there were \(N_{0}\) of these weapons at the outbreak of the war \((t=0),\) then how many were there at time \(t ?\) If the war is a lengthy one, about how many of these weapons will be on hand at time \(t\) when \(t\) is large?
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