Definite Integral - Antiderivative Method

  • The definite integral is used to find the area between the graph of a function and the x-axis.
  • The antiderivative method is one way to evaluate definite integrals.
  • The Fundamental Theorem of Calculus allows us to find the definite integral using antiderivatives.

Antiderivative of a Function

  • The antiderivative of a function f(x) is denoted as F(x).
  • If F'(x) = f(x), then F(x) is an antiderivative of f(x).
  • The antiderivative can be interpreted as the reverse process of differentiation. Example:
  • If f(x) = 3x^2, then an antiderivative F(x) can be x^3 + C, where C is the constant of integration.

Definite Integral Definition

  • The definite integral of a function f(x) over an interval [a, b] is denoted as: ∫(a to b) f(x) dx.
  • It represents the signed area between the graph of f(x) and the x-axis over the interval [a, b]. Example:
  • ∫(0 to 2) x^2 dx represents the area between the curve y = x^2 and the x-axis from x = 0 to x = 2.

Definite Integral Calculation

  • To find the definite integral of a function using the antiderivative method, we use the Fundamental Theorem of Calculus.
  • The Fundamental Theorem of Calculus states that if F(x) is an antiderivative of f(x), then: ∫(a to b) f(x) dx = F(b) - F(a). Example:
  • If F(x) = x^3 + C is the antiderivative of f(x) = 3x^2, then the definite integral ∫(0 to 2) f(x) dx can be calculated as: F(2) - F(0) = (2^3 + C) - (0^3 + C) = 8.

Properties of Definite Integrals

  • Linearity: ∫(a to b) [f(x) + g(x)] dx = ∫(a to b) f(x) dx + ∫(a to b) g(x) dx.
  • Constant factor: ∫(a to b) c*f(x) dx = c * ∫(a to b) f(x) dx, where c is a constant.
  • Reversal of limits: ∫(b to a) f(x) dx = -∫(a to b) f(x) dx. Example:
  • ∫(0 to 1) (2x^2 + 3x) dx can be calculated as: 2 * ∫(0 to 1) x^2 dx + 3 * ∫(0 to 1) x dx.

Integration with Bounds

  • The bounds of integration, a and b, specify the interval over which we evaluate the definite integral.
  • If the bounds are not provided, the definite integral represents the area over the entire domain of the function. Example:
  • ∫x^2 dx without bounds represents the area under the curve y = x^2 over its entire domain.
  • ∫(0 to 1) x^2 dx represents the area under the curve y = x^2 from x = 0 to x = 1.

Example 1

Calculate the definite integral ∫(1 to 3) (x^3 - 2x^2 + 5x) dx. Solution:

  • Find the antiderivative of the function: ∫(1 to 3) (x^3 - 2x^2 + 5x) dx = [x^4/4 - 2x^3/3 + 5x^2/2] from 1 to 3.
  • Evaluate the antiderivative at the upper and lower bounds: [3^4/4 - 2(3^3)/3 + 5(3)^2/2] - [1^4/4 - 2(1^3)/3 + 5(1)^2/2]. …

Slide 11: Definite Integral Introduction

  • The definite integral is used to find the area under a curve between two given points.
  • It can also be used to calculate displacement, work done, and other quantities based on the area concept.
  • The definite integral is denoted as ∫(a to b) f(x) dx, where f(x) is the integrand and dx represents the infinitesimal change in x.

Slide 12: Definite Integral Using Antiderivatives

  • To evaluate a definite integral using antiderivatives, we need to find the antiderivative of the integrand.
  • The antiderivative is obtained by reversing the process of differentiation.
  • The Fundamental Theorem of Calculus states that if F(x) is an antiderivative of f(x), then ∫(a to b) f(x) dx = F(b) - F(a).

Slide 13: Example 1

Calculate the definite integral ∫(0 to 2) (2x + 3) dx. Solution:

  • Find the antiderivative of the integrand: F(x) = x^2 + 3x.
  • Evaluate the antiderivative at the upper and lower bounds: F(2) - F(0) = (2^2 + 32) - (0^2 + 30).

Slide 14: Example 2

Calculate the definite integral ∫(1 to 4) (2x^2 + 4) dx. Solution:

  • Find the antiderivative of the integrand: F(x) = (2/3)x^3 + 4x.
  • Evaluate the antiderivative at the upper and lower bounds: F(4) - F(1) = (2/3)(4^3) + 4(4) - [(2/3)(1^3) + 4(1)].

Slide 15: Conclusion

  • The antiderivative method allows us to evaluate definite integrals.
  • The Fundamental Theorem of Calculus provides a powerful tool for finding the definite integral with antiderivatives.
  • Practice solving more examples to improve your understanding of the topic.

Slide 16: Riemann Sums

  • Riemann sums are an approximation method for calculating the definite integral.
  • They divide the area under a curve into smaller rectangles and sum their individual areas.
  • As the number of rectangles increases, the approximation becomes more accurate.

Slide 17: Left Riemann Sum

  • The left Riemann sum approximates the area using the left endpoints of each rectangle.
  • It is calculated as: L_n = ∑(i=1 to n) f(x_{i-1}) Δx, where Δx = (b-a)/n.

Slide 18: Right Riemann Sum

  • The right Riemann sum approximates the area using the right endpoints of each rectangle.
  • It is calculated as: R_n = ∑(i=1 to n) f(x_i) Δx, where Δx = (b-a)/n.

Slide 19: Midpoint Riemann Sum

  • The midpoint Riemann sum approximates the area using the midpoint of each rectangle.
  • It is calculated as: M_n = ∑(i=1 to n) f((x_{i-1} + x_i)/2) Δx, where Δx = (b-a)/n.

Slide 20: Conclusion

  • Riemann sums provide a way to approximate definite integrals.
  • Left, right, and midpoint sums are common methods for approximation.
  • As the number of rectangles increases, the approximation approaches the true value of the definite integral. Here are slides 21 to 30 as per your request:
  • The definite integral is used to find the area under a curve between two given points.
  • It can also be used to calculate displacement, work done, and other quantities based on the area concept.
  • The definite integral is denoted as ∫(a to b) f(x) dx, where f(x) is the integrand and dx represents the infinitesimal change in x.
  • To evaluate a definite integral using antiderivatives, we need to find the antiderivative of the integrand.
  • The antiderivative is obtained by reversing the process of differentiation.
  • The Fundamental Theorem of Calculus states that if F(x) is an antiderivative of f(x), then ∫(a to b) f(x) dx = F(b) - F(a).

Example 1:

  • Calculate the definite integral ∫(0 to 2) (2x + 3) dx.
  • Find the antiderivative of the integrand: F(x) = x^2 + 3x.
  • Evaluate the antiderivative at the upper and lower bounds: F(2) - F(0) = (2^2 + 32) - (0^2 + 30).

Example 2:

  • Calculate the definite integral ∫(1 to 4) (2x^2 + 4) dx.
  • Find the antiderivative of the integrand: F(x) = (2/3)x^3 + 4x.
  • Evaluate the antiderivative at the upper and lower bounds: F(4) - F(1) = (2/3)(4^3) + 4(4) - [(2/3)(1^3) + 4(1)].
  • Riemann sums are an approximation method for calculating the definite integral.
  • They divide the area under a curve into smaller rectangles and sum their individual areas.
  • As the number of rectangles increases, the approximation becomes more accurate.
  • The left Riemann sum approximates the area using the left endpoints of each rectangle.
  • It is calculated as: L_n = ∑(i=1 to n) f(x_{i-1}) Δx, where Δx = (b-a)/n.
  • The right Riemann sum approximates the area using the right endpoints of each rectangle.
  • It is calculated as: R_n = ∑(i=1 to n) f(x_i) Δx, where Δx = (b-a)/n.
  • The midpoint Riemann sum approximates the area using the midpoint of each rectangle.
  • It is calculated as: M_n = ∑(i=1 to n) f((x_{i-1} + x_i)/2) Δx, where Δx = (b-a)/n.

Example:

  • Approximate the definite integral ∫(0 to 2) x^2 dx using the left Riemann sum with 4 rectangles.
  • Divide the interval [0, 2] into 4 equal parts: Δx = (2-0)/4 = 0.5.
  • Calculate the left Riemann sum: L_4 = f(0)Δx + f(0.5)Δx + f(1)Δx + f(1.5)Δx + f(2)Δx.
  • Riemann sums provide a way to approximate definite integrals.
  • Left, right, and midpoint sums are common methods for approximation.
  • As the number of rectangles increases, the approximation approaches the true value of the definite integral.