Slide 1
Title: Integral Calculus - Intersection of a curve with another curve
Slide 2
Introduction to intersection of curves in integral calculus
Definition: Intersection point(s) occur when two curves intersect each other
The coordinates of intersection point(s) can be found using integration
Slide 3
Example 1:
Find the intersection point(s) of the curves y = x^2 and y = 2x - 1
Slide 4
Solution to Example 1:
Set the equations equal to each other:
x^2 = 2x - 1
Rearrange to form a quadratic equation:
x^2 - 2x + 1 = 0
Solve the quadratic equation to find the values of x
Substitute the values of x into one of the equations to find the corresponding y values
Slide 5
Example 2:
Find the intersection point(s) of the curves y = sin(x) and y = cos(x)
Slide 6
Solution to Example 2:
Set the equations equal to each other:
sin(x) = cos(x)
Rearrange using trigonometric identities:
tan(x) = 1
Solve for x by taking the inverse tangent of both sides
Slide 7
Example 3:
Find the intersection point(s) of the curves y = e^x and y = ln(x)
Slide 8
Solution to Example 3:
Set the equations equal to each other:
e^x = ln(x)
Solve for x numerically using a graphing calculator or algebraic methods
Substitute the values of x into one of the equations to find the corresponding y values
Slide 9
Intersection of curves in the context of area under a curve
Finding the area between two curves using integration
Slide 10
Example 4:
Find the area between the curves y = x^2 and y = 2x - 1
Slide 11
Solution to Example 4:
Set the equations equal to each other:
x^2 = 2x - 1
Rearrange to form a quadratic equation:
x^2 - 2x + 1 = 0
Solve the quadratic equation to find the x values
Determine the limits of integration based on the intersection points
Use the definite integral to find the area between the curves
Slide 12
Example 5:
Find the area between the curves y = sin(x) and y = cos(x) in the interval [0, π/2]
Slide 13
Solution to Example 5:
Set the equations equal to each other:
sin(x) = cos(x)
Rearrange using trigonometric identities:
tan(x) = 1
Solve for x by taking the inverse tangent of both sides
Determine the limits of integration based on the intersection points
Use the definite integral to find the area between the curves
Slide 14
Example 6:
Find the area between the curves y = e^x and y = ln(x) in the interval [1, e]
Slide 15
Solution to Example 6:
Set the equations equal to each other:
e^x = ln(x)
Solve for x numerically using a graphing calculator or algebraic methods
Determine the limits of integration based on the intersection points
Use the definite integral to find the area between the curves
Slide 16
Intersection of curves in the context of finding the common region
Finding the common region between two curves using integration
Slide 17
Example 7:
Find the common region of the curves y = x and y = x^2
Slide 18
Solution to Example 7:
Set the equations equal to each other:
x = x^2
Rearrange to form a quadratic equation:
x^2 - x = 0
Solve the quadratic equation to find the x values
Determine the limits of integration based on the intersection points
Use the definite integral to find the area of the common region
Slide 19
Example 8:
Find the common region of the curves y = sin(x) and y = cos(x) in the interval [0, 2π]
Slide 20
Solution to Example 8:
Set the equations equal to each other:
sin(x) = cos(x)
Rearrange using trigonometric identities:
tan(x) = 1
Solve for x by taking the inverse tangent of both sides
Determine the limits of integration based on the intersection points
Use the definite integral to find the area of the common region
Slide 21
Example 9:
Find the common region of the curves y = e^x and y = ln(x) in the interval [1, e]
Slide 22
Solution to Example 9:
Set the equations equal to each other:
e^x = ln(x)
Solve for x numerically using a graphing calculator or algebraic methods
Determine the limits of integration based on the intersection points
Use the definite integral to find the area of the common region
Slide 23
Summary of finding intersection points:
Set the equations equal to each other
Solve the resulting equation(s) for x
Substitute the x values into one of the equations to find the corresponding y values
Slide 24
Summary of finding area between curves:
Find the intersection points of the curves
Determine the limits of integration based on the intersection points
Use the definite integral to find the area between the curves
Slide 25
Summary of finding common region:
Find the intersection points of the curves
Determine the limits of integration based on the intersection points
Use the definite integral to find the area of the common region
Slide 26
Key takeaways:
Intersection of curves is determined by finding the values of x and y where two curves meet
The area between curves is found by integrating the difference between the two curves over the given interval
The common region between curves is found by integrating the minimum or maximum of the two curves over the given interval
Slide 27
Summary of concepts covered:
Definition of intersection of curves
Finding intersection points algebraically
Finding intersection points numerically
Finding area between curves
Finding common region between curves
Slide 28
Additional resources:
Textbooks:
“Calculus” by James Stewart
“Advanced Engineering Mathematics” by Erwin Kreyszig
Online resources:
Khan Academy:
www.khanacademy.org
Wolfram Alpha:
www.wolframalpha.com
Slide 29
Practice exercises:
Find the intersection points of the curves y = x^3 and y = x^2
Find the area between the curves y = cos(x) and y = sin(x) in the interval [0, π]
Find the common region of the curves y = e^x and y = 1/x in the interval [1, e]
Slide 30
Any questions or doubts?
Thank you for your attention!