Definite Integral - Example Of Integration- An Introduction

Definite Integral - Example of Integration: An Introduction

Definition of definite integral
Different notations for definite integral
Importance and applications of definite integral
Connection between definite integral and area under a curve
The fundamental theorem of calculus

Definition of Definite Integral

The definite integral represents the signed area between the curve and the x-axis in a given interval.
denoted by symbol ∫
If f(x) is a continuous function on [a, b], then the definite integral of f(x) from a to b is given by: ∫(from a to b) f(x) dx

Notations for Definite Integral

The integral can be represented using different notations:
- ∫(from a to b) f(x) dx
- ∫ a^b f(x) dx
- ∫ f(x)|_(a to b)

Importance and Applications of Definite Integral

Calculation of areas and volumes
Calculation of displacement and distance
Evaluation of definite integrals helps in solving various physics problems
Calculation of average value of a function

Connection between Definite Integral and Area under a Curve

The definite integral of a function f(x) from a to b represents the area under the curve between a and b.
The area can be positive or negative based on the behavior of the function.
The area under the curve can be calculated by evaluating the definite integral.

The Fundamental Theorem of Calculus

The fundamental theorem of calculus states that:
- If F(x) is an antiderivative of a function f(x) on an interval [a, b], then: ∫(from a to b) f(x) dx = F(b) - F(a)

Example 1: Finding the Area of a Region

Find the area of the region bounded by the curve y = x^2 - 3x + 2, the x-axis, and the lines x = 0 and x = 2.
Solution: We can find the area by evaluating the definite integral of the function from 0 to 2: Area = ∫(from 0 to 2) (x^2 - 3x + 2) dx

Example 2: Calculating Displacement

A particle is moving along the x-axis with a velocity given by v(t) = 3t^2 - 2t + 1. Find the displacement of the particle from time t = 0 to t = 3.
Solution: Displacement can be found by evaluating the definite integral of the velocity function from 0 to 3: Displacement = ∫(from 0 to 3) (3t^2 - 2t + 1) dt

Example 3: Average Value of a Function

Find the average value of the function f(x) = 2x^3 - x^2 + 3x - 1 on the interval [0, 4].
Solution: The average value of a function can be found by evaluating the definite integral of the function over the interval and dividing it by the length of the interval: Average value = (1/4) * ∫(from 0 to 4) (2x^3 - x^2 + 3x - 1) dx

Summary

Definite integral represents the signed area between the curve and the x-axis in a given interval.
Notations for definite integral: ∫(from a to b) f(x) dx, ∫ a^b f(x) dx, ∫ f(x)|_(a to b)
Definite integral has applications in finding areas, volumes, displacement, distance, and average value of a function.
The fundamental theorem of calculus connects definite integrals and antiderivatives.
Examples: Finding area, calculating displacement, and finding average value of a function.

Properties of Definite Integral

Linearity property: ∫(from a to b) (cf(x) + dg(x)) dx = c*∫(from a to b) f(x) dx + d*∫(from a to b) g(x) dx
Additivity property: ∫(from a to c) f(x) dx + ∫(from c to b) f(x) dx = ∫(from a to b) f(x) dx
Change of limits property: ∫(from a to b) f(x) dx = -∫(from b to a) f(x) dx

Methods of Definite Integration

Integration by substitution
Integration by parts
Integration by partial fractions
Using trigonometric identities
Special integration techniques: trigonometric substitution, integration tables, and numerical methods

Example 4: Integration by Substitution

Evaluate the definite integral ∫(from 0 to 1) 2x(e^x^2) dx using the method of integration by substitution.
Solution: Let u = x^2, then du = 2x dx. Substituting these values, the integral becomes: ∫(from u=0 to u=1) e^u du

Example 5: Integration by Parts

Evaluate the definite integral ∫(from 0 to 1) x*sin(x) dx using the method of integration by parts.
Solution: Using the formula for integration by parts, choose u = x and dv = sin(x) dx. Differentiating and integrating, the integral becomes: ∫(from 0 to 1) -x*cos(x) dx

Example 6: Integration by Partial Fractions

Evaluate the definite integral ∫(from 0 to 4) (x^2 + 3x + 2)/(x^3 + 6x^2 + 11x + 6) dx using the method of integration by partial fractions.
Solution: Factoring the denominator and decomposing the rational function, the integral becomes: ∫(from 0 to 4) (1/(x+1)) + (1/(x+2)) dx

Example 7: Using Trigonometric Identities

Evaluate the definite integral ∫(from 0 to π/2) sin^3(x) dx using trigonometric identities.
Solution: Using the identity sin^3(x) = (3sin(x) - sin(3x))/4, the integral becomes: ∫(from 0 to π/2) (3sin(x) - sin(3x))/4 dx

Example 8: Trigonometric Substitution

Evaluate the definite integral ∫(from 0 to 1) dx/(1 + x^2)^0.5 using trigonometric substitution.
Solution: Substitute x = tan(t), then dx = sec^2(t) dt. The integral becomes: ∫(from 0 to π/4) sec(t) dt

Example 9: Using Integration Tables

Evaluate the definite integral ∫(from 0 to 1) e^x/x dx using integration tables.
Solution: The integral does not have an elementary antiderivative. However, using integration tables, we can find the integral as: ∫(from 0 to 1) Ei(x) dx

Example 10: Numerical Integration

Approximate the definite integral ∫(from 0 to 1) e^(-x^2) dx using the trapezoidal rule with 4 subintervals.
Solution: Divide the interval [0, 1] into 4 equal subintervals and approximate the integral using the trapezoidal rule formula: ((1/8)*(e^0 + 2e^(-1/16) + 2e^(-1/4) + e^(-1/2)))

Summary

Properties of definite integral: linearity, additivity, and change of limits.
Methods of definite integration: substitution, parts, partial fractions, trigonometric identities, special techniques, and numerical methods.
Examples: Integration by substitution, parts, partial fractions, using trigonometric identities, trigonometric substitution, integration tables, and numerical integration. ``markdown

Examples

Example 11: Finding the area between two curves
- Given two functions f(x) and g(x), find the area between the curves in a specified interval.
- Solution: The area can be calculated by finding the difference between the definite integrals of the two functions.
Example 12: Solving physics problems involving definite integrals
- Use definite integrals to solve problems related to motion, work, and other physical quantities.
- Solution: Express the problem in terms of a function and use the definite integral to find the desired quantity.
Example 13: Evaluating definite integrals involving transcendental functions
- Evaluate definite integrals involving functions such as exponential, logarithmic, and trigonometric functions.
- Solution: Use appropriate substitution or integration techniques to simplify the integral and evaluate it.

Techniques for Definite Integral Calculations

Riemann sums and rectangular approximation
Optimization problems
Improper integrals
Numerical methods such as Simpson’s rule and Gaussian quadrature

Example 14: Riemann Sums and Rectangular Approximation

Estimate the value of the definite integral ∫(from 0 to 4) x^2 dx using 4 subintervals and left endpoints of the intervals for approximation.
Solution: Divide the interval [0, 4] into 4 subintervals and use the left endpoints of the intervals to calculate the Riemann sum.

Example 15: Optimization Problems

Find the dimensions of a rectangular box with a fixed surface area of 100 square units that minimize the volume.
Solution: Express the volume and surface area in terms of a single variable, apply the necessary calculus techniques to optimize the value.

Example 16: Improper Integrals

Evaluate the improper integral ∫(from 1 to ∞) 1/x dx.
Solution: The integral is improper because the upper limit is infinite. Evaluate the integral as the limit of definite integrals.

Example 17: Simpson’s Rule

Approximate the definite integral ∫(from 0 to π/2) sin(x) dx using Simpson’s rule with 6 subintervals.
Solution: Apply Simpson’s rule formula to calculate the approximate value of the integral using the given number of subintervals.

Example 18: Gaussian Quadrature

Approximate the definite integral ∫(from -1 to 1) e^(-x^2) dx using Gaussian quadrature with 3 points.
Solution: Use the weights and nodes for Gaussian quadrature to calculate the approximate value of the integral.

Summary

Further examples: Finding the area between two curves, solving physics problems, and evaluating integrals involving transcendental functions.
Techniques for definite integral calculations: Riemann sums, optimization problems, improper integrals, and numerical methods such as Simpson’s rule and Gaussian quadrature.
Examples: Using Riemann sums, solving optimization problems, evaluating improper integrals, and using Simpson’s rule and Gaussian quadrature.

Conclusion

Definite integrals are a fundamental concept in calculus and have various applications in mathematics and other fields.
Understanding and proficiently solving problems involving definite integrals is crucial for success in the mathematics examination.
Practice solving a variety of problems and use different techniques to strengthen your understanding and problem-solving skills.
Review the techniques discussed in this lecture, and apply them to new problems to improve your problem-solving abilities.

Questions

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