Fundamental Theorem Of Definite Integrals

Unit: Integration & Accumulation of Change

Chapter: Fundamental Theorem of Definite Integral

Reference: – Antiderivatives, Indefinite integrals, Definite integrals, Fundamental theorem of Calculus, Part 1 & Part 2 of FTC, Evaluation using Antiderivatives, Reverse power rule, Initial value problem, Area under the curve, Mean value theorem & Properties, Applications.

 

After studying this chapter, you should be able to:

  • Introduction to Definite Integral & Accumulation Functions.
  • Evaluation of definite Integral & Antiderivatives.
  • Indefinite Integrals methods.

 

Introduction to Definite Integrals

 

  1. Bounds: The definite integral is evaluated over a specified interval, denoted as [a, b]. The values 'a' and 'b' are called the lower and upper bounds, respectively. The integral calculates the accumulated change of the function within this interval.
  2. Net Area: The definite integral represents the net signed area between the function and the x-axis over the interval [a, b]. Positive areas above the x-axis contribute positively to the integral, while negative areas below the x-axis contribute negatively.
  3. Notation: The definite integral is denoted using the integral symbol (∫), followed by the function to be integrated and the differential variable. For example, if we want to calculate the definite integral of a function f(x) over the interval [a, b], it is written as ∫[a to b] f(x) dx.
  4. Evaluation: To evaluate a definite integral, you need to find an antiderivative (also known as an indefinite integral or primitive) of the integrand. Once you have the antiderivative, you substitute the upper limit (b) into the antiderivative, subtract the result of substituting the lower limit (a), and simplify the expression.
  5. Accumulation of Change: The definite integral represents the accumulation of change in a function over the interval [a, b]. It measures the net change in the function's values between 'a' and 'b'. This accumulated change can represent physical quantities such as displacement, total distance traveled, or total area under a curve.
  6. Connection to the Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus relates the definite integral to the antiderivative of a function. It states that if F(x) is an antiderivative of a continuous function f(x), then the definite integral of f(x) from 'a' to 'b' is equal to F(b) – F(a).

 

Accumulation Functions:

  1. Definition: An accumulation function, also known as an antiderivative or indefinite integral, is the reverse process of differentiation. Given a function f(x), an accumulation function, denoted by F(x), is a function whose derivative is equal to f(x). In other words, F'(x) = f(x).
  2. Relationship to Definite Integral: The accumulation function is closely tied to the definite integral. If F(x) is an accumulation function of a function f(x), then the definite integral of f(x) from 'a' to 'b' can be calculated as the difference between the values of F(x) at 'b' and 'a': ∫[a to b] f(x) dx = F(b) – F(a).
  3. Interpretation: The accumulation function represents the accumulated change in a function over a given interval. It measures the net area under the curve of the function from some fixed reference point to a particular value of x. It can also represent quantities such as total distance traveled or total accumulated value.
  4. A constant of Integration: When finding an accumulation function, it is important to include a constant of integration, typically denoted as 'C'. Since the derivative of a constant is zero, any constant added to an accumulation function will also have a derivative of zero. This constant accounts for all possible antiderivatives of f(x) that differ by a constant.
  5. Calculation: To find an accumulation function, you need to determine the antiderivative of the given function. This involves finding a function whose derivative matches the given function. The antiderivative can be found using techniques such as basic rules of integration, integration by substitution, integration by parts, or tables of common antiderivatives.
  6. Example: Let's consider the function f(x) = 2x. To find its accumulation function F(x), we can integrate f(x) for x: ∫ 2x dx = x2 + C, where C represents the constant of integration. Therefore, the accumulation function F(x) for f(x) = 2x is F(x) = x2 + C

Partitioning the Interval & Riemann sum Notation: –

Partitioning the Interval:

  1. An interval is a range of values on the number line, often denoted as [a, b], where "a" represents the lower limit and "b" represents the upper limit.
  2. Partitioning the interval involves dividing it into smaller subintervals or intervals.
  3. The process of partitioning determines the width of each subinterval, which affects the accuracy of the Riemann sum approximation.
  4. The partitioning can be done by choosing specific points within the interval to create the subintervals.

Riemann Sum Notation:

  1. Riemann sums are typically represented using sigma (Σ) notation, which is a shorthand way of expressing a sum.
  2. The sigma symbol Σ is followed by the variable that represents the index of summation. This variable is often denoted as "i" or "k".
  3. The lower and upper limits of summation are written as subscripts below and above the sigma symbol, respectively. These limits define the range of values for the index variable.
  4. The expression inside the sigma symbol represents the function evaluated at specific points.
  5. Riemann sum notation combines the function evaluation with the width of the subintervals to calculate the areas of the corresponding rectangles.
  6. Depending on the type of Riemann sum, the expression inside the sigma symbol may involve using left, right, or midpoint endpoints of the subintervals to determine the height of the rectangles.

To put it all together, when working with Riemann sums:

  1. Partition the interval by dividing it into smaller subintervals.
  2. Determine the width of each subinterval.
  3. Choose the appropriate Riemann sum type (left, right, or midpoint).
  4. Use Riemann sum notation with sigma notation to express the approximation.
  5. Calculate the height of each rectangle using the chosen Riemann sum method.
  6. Multiply the height of each rectangle by its width to obtain the area.
  7. Sum up all the areas of the rectangles to approximate the total area under the curve.

 

Approximation Area & Rectangular Approximation Method:

Approximation of Area:

 

  1. In calculus, the problem of finding the exact area under a curve can be challenging. To tackle this, approximation methods are used to estimate the area.
  2. The idea is to divide the region under the curve into smaller, simpler shapes like rectangles and then calculate the sum of their areas.
  3. The sum of these areas provides an approximation of the total area under the curve.
  4. By using more narrower rectangles, the approximation becomes more accurate.

Rectangular Approximation Method:

 

  • Rectangular approximation is one of the simplest methods for estimating the area under a curve.
  • The region under the curve is divided into a series of rectangles, and the sum of their areas is calculated.
  • The width of each rectangle is determined by partitioning the interval of integration.
  • The height of each rectangle is determined based on the function being integrated.
  • The choice of endpoints (left, right, or midpoint) for determining the height of the rectangles depends on the specific Riemann sum being used.
  • The left endpoint rule (or lower sum) uses the left endpoint of each subinterval to determine the rectangle height.
  • The right endpoint rule (or upper sum) uses the right endpoint of each subinterval to determine the rectangle height.
  • The midpoint rule uses the midpoint of each subinterval to determine the rectangle height.
  • Once the heights and widths of the rectangles are determined, the areas of each rectangle are calculated by multiplying the height by the width.
  • The areas of all the rectangles are then summed to obtain an approximation of the area under the curve.

Example: – Find the value of the definite integral ∫[0 to 3] (2x + 1) dx.

 

Solution: To evaluate the definite integral, we can apply the Fundamental Theorem of Definite Integration. First, we need to find an antiderivative of the integrand, which is (2x + 1).

 

Step 1: Find the antiderivative:

To find an antiderivative of 2x + 1, we integrate term by term. The antiderivative of 2x is x2, and the antiderivative of 1 is x:

∫ (2x + 1) dx = ∫ 2x dx + ∫ 1 dx = x2 + x + C.

Step 2: Evaluate the definite integral:

Now, we can evaluate the definite integral by subtracting the value of the antiderivative at the lower limit (0) from its value at the upper limit (3):

∫[0 to 3] (2x + 1) dx = (x2 + x + C) |[0 to 3].

Plugging in the upper limit (3) into the antiderivative, we have:

(x2 + x + C) |[0 to 3] = (32 + 3 + C) – (02 + 0 + C) = (9 + 3 + C) – (0 + 0 + C) = 12 + C – C = 12.

Therefore, the value of the definite integral ∫[0 to 3] (2x + 1) dx is 12.

Example 2:   Evaluate the definite integral   (3x2 + 2x) dx.

 

Solution: To evaluate the definite integral, we can apply the Fundamental Theorem of Definite Integration. First, we need to find an antiderivative of the integrand, which is (3x2 + 2x).

 

Step 1: Find the antiderivative:

To find an antiderivative of 3x^2 + 2x, we integrate term by term. The antiderivative of 3x2 is x3, and the antiderivative of 2x is x2:

∫ (3x2 + 2x) dx = ∫ 3x2 dx + ∫ 2x dx = x3 + x2 + C.

 

Step 2: Evaluate the definite integral:

Now, we can evaluate the definite integral by subtracting the value of the antiderivative at the lower limit (1) from its value at the upper limit (4):

 (3x2 + 2x) dx = (x3 + x2 + C) |.

 

Plugging in the upper limit (4) into the antiderivative, we have:

= (64 + 16 + C) – (1 + 1 + C) = 80 + C – 2 – C = 78.

 

Therefore, the value of the definite integral   (3x2 + 2x) dx is 78.

Key Points

  • The Fundamental Theorem of Definite Integration establishes a connection between differentiation and integration.

 

  • It consists of two parts: FTC – Part I and FTC – Part II.

 

  • FTC – Part I states that if a function f(x) is continuous on an interval [a, b] and F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b is equal to the difference between F(b) and F(a).

 

  • FTC – Part II provides a method for evaluating definite integrals. It states that if F(x) is an antiderivative of a continuous function f(x) on the interval [a, b], then the definite integral of f(x) from a to b can be evaluated by subtracting the value of F(x) at the lower limit from its value at the upper limit.

 

  • The Fundamental Theorem of Definite Integration connects the concept of accumulation of change to definite integrals.

 

  • The definite integral represents the net area between the function and the x-axis over a given interval.

 

  • The definite integral can be used to calculate quantities such as area, displacement, and accumulated value.

 

  • An accumulation function (antiderivative) represents the accumulated change in a function over a specific interval.

 

  • The definite integral is equal to the difference between the values of the accumulation function at the upper and lower limits of integration.

 

  • The constant of integration is included when finding the accumulation function.

 

  • The Fundamental Theorem of Definite Integration is crucial for solving problems involving definite integrals, such as finding areas under curves or calculating total accumulated quantities.

 

  • It provides a powerful tool for connecting the concepts of differentiation and integration, allowing for efficient calculations and interpretations in calculus.

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