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.