Calculus Integration Techniques
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Questions and Answers

What is the significance of using part (b) of Definition (4–2) in determining the improper integral's convergence in the provided text?

  • It means the function is unbounded at the left endpoint of the interval. (correct)
  • It indicates a vertical asymptote at the right endpoint of the interval.
  • It suggests a vertical asymptote at the left endpoint of the interval.
  • It implies the function is unbounded at the right endpoint of the interval.

    • Based on the discussion in the text, how is the value of an improper integral interpreted when the integrand is positive?

  • As the area under the curve, limited by the x-axis and the function. (correct)
  • As the slope of the function at a point within the interval.
  • As the average value of the function over the interval.
  • As a point of inflection on the graph of the function.

    • What is implied by a function having a vertical asymptote at a specific point within an interval, as discussed in the text?

  • The function is continuous at that point.
  • The integral converges at that point.
  • The integral diverges at that point. (correct)
  • The function is bounded at that point.

    • In evaluating improper integrals, what does using l'Hopital’s Rule help determine?

    <p>The limit of functions involving infinity.</p> Signup and view all the answers

    What is an improper integral according to the text?

    <p>An integral with an infinite discontinuity</p> Signup and view all the answers

    In the context of the text, what type of region is considered for improper integrals in Type 1?

    <p>An infinite region under a curve</p> Signup and view all the answers

    How is the area of the region S, under the curve y = 1/x^2 and to the right of x = 1, interpreted in the text?

    <p>The area is equal to 1</p> Signup and view all the answers

    How does the text define the integral of a function over an infinite interval?

    <p>As the limit of integrals over finite intervals</p> Signup and view all the answers

    For which values of $p$ is the integral $ \int_{1}^{\infty} \frac{1}{x^p} dx$ convergent?

    <p>$p &gt; 1$</p> Signup and view all the answers

    In Example (4 - 3), what is the interpretation of the improper integral $ \int_{2}^{\infty} \frac{1}{x} dx$?

    <p>Area under the curve $y = \frac{1}{x}$ above the x-axis</p> Signup and view all the answers

    In the context of integrating by parts, if $u = x$ and $dv = e^x dx$, what is the value of $ du$?

    <p>$1$</p> Signup and view all the answers

    Which of the following statements is true regarding the Type 2 improper integrals mentioned in the text?

    <p>$S$ is unbounded between $a$ and $b$</p> Signup and view all the answers

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