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Introduction to Definite Integrals quiz #1

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  • What is a definite integral and how does it relate to the area under a curve?
    A definite integral, written as ∫ₐᵇ f(x) dx, represents the exact area under the curve of f(x) from x = a to x = b by taking the limit of Riemann sums as the number of rectangles approaches infinity.
  • How does increasing the number of rectangles in a Riemann sum affect the approximation of the area under a curve?
    Increasing the number of rectangles in a Riemann sum makes the approximation more accurate, and as the number approaches infinity, the sum becomes the exact area, which is the definite integral.
  • What is the sum and difference rule for definite integrals?
    The sum and difference rule states that the integral of a sum or difference of functions equals the sum or difference of their integrals: ∫ₐᵇ [f(x) ± g(x)] dx = ∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx.
  • What does the constant multiple rule state for definite integrals?
    The constant multiple rule states that a constant can be factored out of the integral: ∫ₐᵇ k·f(x) dx = k·∫ₐᵇ f(x) dx.
  • How does changing the order of the bounds in a definite integral affect its value?
    Switching the bounds of integration reverses the sign of the integral: ∫ₐᵇ f(x) dx = -∫ᵦₐ f(x) dx.
  • What is the value of a definite integral when the upper and lower bounds are the same?
    The value is zero: ∫ₐₐ f(x) dx = 0, because there is no interval and thus no area under the curve.
  • Explain the additivity rule for definite integrals.
    The additivity rule states that if c is between a and b, then ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx = ∫ₐᵇ f(x) dx.
  • How do you set up a definite integral to find the area under the curve of f(x) from x = 2 to x = 5?
    Set up the integral as ∫₂⁵ f(x) dx, which represents the area under f(x) from x = 2 to x = 5.
  • If you have ∫₀⁴ (x + 1) dx, what does this integral represent?
    It represents the exact area under the curve y = x + 1 from x = 0 to x = 4.
  • Why does integrating from a to a always result in zero area?
    Because there is no width between the bounds, so the 'area' under the curve is zero.
  • What is the main difference between a Riemann sum and a definite integral?
    A Riemann sum approximates the area under a curve using a finite number of rectangles, while a definite integral gives the exact area by taking the limit as the number of rectangles approaches infinity.
  • How do you evaluate the definite integral of the function f(x) = 1 + 7x from x = 0 to x = 3?
    To evaluate the definite integral ∫₀³ (1 + 7x) dx, first find the antiderivative: ∫(1 + 7x) dx = x + (7/2)x². Then, apply the Fundamental Theorem of Calculus by substituting the upper and lower bounds: [x + (7/2)x²] from 0 to 3. Calculate the value at x = 3 and subtract the value at x = 0: (3 + (7/2)(3)²) - (0 + (7/2)(0)²) = 3 + (7/2)(9) = 3 + 31.5 = 34.5.