{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

03Integration with a Boundary Condition and The Definite Integral2

# 03Integration with a Boundary Condition and The Definite Integral2

This preview shows pages 1–4. Sign up to view the full content.

I NTEGRATION WITH A B OUNDARY C ONDITION AND T HE D EFINITE I NTEGRAL Initial Value Problems When we integrate a function ) ( x f we get a family of functions C x F + ) ( , but sometimes we are interested in one particular function. We can identify this function by specifying a specific point it passes through. Ex . 2 2 3 ) ( x x x f - + = 1 ) 0 ( = F This question can also be presented as 2 2 3 x x dx dy - + = 1 ) 0 ( = y Where the unknown is not a number, but a function. This type of problem (involving the derivative of an unknown function) is called a differential equation. The Definite Integral Area Interpretation positive area 1 A 3 A a 2 A b negative area Total Area = Area over x -axis – Area under x -axis = 2 3 1 A A A - +

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Notation : b a dx x f ) ( is the definite integral of f from a to b a is the lower limit, b is the upper limit, ) ( x f is the integrand Ex . Sketch and evaluate using geometric formulae. 1) 4 1 2 dx 2) - + 2 1 ) 2 ( dx x 3) - 2 0 ) 1 ( dx x Note: Must split into two integrals. (Total area is 0). ) ( ) ( x f x A = A 0 ) ( = a A = b a dx x f A ) ( A b A = ) ( a b ) ( ) ( x f x A = let C x A x F + = ) ( ) ( Area = [ ] [ ] A A a A b A c a A c b A a F b F = - = - = + - + = - 0 ) ( ) ( ) ( ) ( ) ( ) ( So, ) ( ) ( ) ( a F b F dx x f b a - =
The Fundamental Theorem of Calculus Part 1: If f if continuous on

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern