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Chapter 5: Problem 10

In Exercises \(7-12,\) evaluate the integral. $$\int_{-4}^{-1} \frac{\pi}{2} d \theta$$

Short Answer

Expert verified
The value of the integral \(\int_{-4}^{-1} \frac{\pi}{2} d \theta\) is \(\frac{3\pi}{2}\)

Step by step solution

01

Identify the function and boundaries

The function to integrate is \(\frac{\pi}{2}\), which is a constant function. The boundaries for the integral are -4 and -1. Therefore, the integral to compute is \(\int_{-4}^{-1} \frac{\pi}{2} d \theta\).
02

Compute the integral of the constant function

Since the function is a constant, we can factor it out of the integral and multiply it by the difference between the upper and lower bounds of the integral. It gives us \(\frac{\pi}{2} 脳 [-1 - (-4)] = \frac{3\pi}{2}\).
03

Conclusion

Therefore, the integral of \(\frac{\pi}{2}\) from -4 to -1 is \(\frac{3\pi}{2}\).

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Key Concepts

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

Constant Function Integration
When you come across an integral with a constant function, you're in luck because it's one of the simplest types to solve. A constant function is a function that remains the same, no matter the input. In the exercise given, the function \(\frac{\pi}{2}\) is a constant.

To integrate a constant, simply multiply the constant by the difference in the upper and lower limits of the integral. For example:
  • Identify the constant (here it's \(\frac{\pi}{2}\)).
  • Determine the limits of integration (from \(-4\) to \(-1\) in this case).
  • Compute the difference in these limits. Here, it's \(-1 - (-4) = 3\).
  • Multiply the constant by the difference: \[ \frac{\pi}{2} \times 3 = \frac{3\pi}{2}. \]
Constant functions make integration straightforward, reducing the problem to basic arithmetic on the limits.
Calculus Problem Solving
Problem-solving in calculus often involves breaking down problems into manageable parts. To tackle an integral, follow a structured approach by identifying the type of integrand - in this instance, a constant function.
Under calculus, several steps are consistently involved:
  • Identify the type of integral you are dealing with: Here, it's decided as an integral of a constant function.
  • Simplify the function if possible: For constants, there really isn't anything to simplify.
  • Solve the integration by applying the defined formula or method: Remember our simple arithmetic multiplication for constants.
  • Ensure you perform calculations on limits accurately: Double-check that subtraction between the bounds is correct.
  • Verify your solution: Especially with subtraction, repeating the calculation ensures no arithmetic errors were made.
Using these steps streamlines calculus problem-solving, embracing the ease with which constants can be integrated.
Integrals with Boundaries
Integrals with boundaries, also called definite integrals, quantify the net area between the curve and the x-axis, from the start of one boundary to the end of another. Boundaries are crucial in determining the specific range over which to evaluate the integral.
In the exercise:
  • The boundaries are \(-4\) to \(-1\), indicating where you start and stop integrating.
  • With definite integrals, the result is a numerical value due to these limits. The integral gives you \[ \frac{3\pi}{2}, \] representing the net area for \( \frac{\pi}{2} \) between these two points.
Always double-check that the limits are input correctly, as they directly influence the result. By evaluating within specified boundaries, integrals help us model and solve real-world problems such as calculating displacements or areas.

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Most popular questions from this chapter

Revenue from Marginal Revenue Suppose that a company's marginal revenue from the manufacture and sale of egg beaters is \(\frac{d r}{d x}=2-\frac{2}{(x+1)^{2}}\)where \(r\) is measured in thousands of dollars and \(x\) in thousands of units. How much money should the company expect from a production run of \(x=3\) thousand eggbeaters? To find out, integrate the marginal revenue from \(x=0\) to $x=3 . \quad \$

In Exercises 55 and \(56,\) find \(K\) so that $$\int_{a}^{x} f(t) d t+K=\int_{b}^{x} f(t) d t$$ $$f(x)=\sin ^{2} x ; \quad a=0 ; \quad b=2$$

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Multiple Choice Let \(f(x)=\int_{a}^{x} \ln (2+\sin t) d t .\) If \(f(3)=4\) then \(f(5)=\) \(\begin{array}{lllll}{\text { (A) } 0.040} & {\text { (B) } 0.272} & {\text { (C) } 0.961} & {\text { (D) } 4.555} & {\text { (E) } 6.667}\end{array}\)

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