Chap6_Sec1 - 6 APPLICATIONS OF INTEGRATION APPLICATIONS...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6
Background image of page 1

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

View Full Document Right Arrow Icon
In this chapter, we explore some of the applications of the definite integral by using it to compute areas between curves, volumes of solids, and the work done by a varying force. The common theme is the following general method— which is similar to the one used to find areas under curves. APPLICATIONS OF INTEGRATION
Background image of page 2
We break up a quantity Q into a large number of small parts. Next, we approximate each small part by a quantity of the form and thus approximate Q by a Riemann sum. Then, we take the limit and express Q as an integral. Finally, we evaluate the integral using the Fundamental Theorem of Calculus or the Midpoint Rule. APPLICATIONS OF INTEGRATION ( *) i f x x
Background image of page 3

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

View Full Document Right Arrow Icon
6.1 Areas Between Curves In this section we learn about: Using integrals to find areas of regions that lie between the graphs of two functions. APPLICATIONS OF INTEGRATION
Background image of page 4
Consider the region S that lies between two curves y = f ( x ) and y = g ( x ) and between the vertical lines x = a and x = b . Here, f and g are continuous functions and f ( x ) ≥ g ( x ) for all x in [ a , b ]. AREAS BETWEEN CURVES
Background image of page 5

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

View Full Document Right Arrow Icon
As we did for areas under curves in Section 5.1, we divide S into n strips of equal width and approximate the i th strip by a rectangle with base ∆ x and height . ( *) ( *) i i f x g x - AREAS BETWEEN CURVES
Background image of page 6
We could also take all the sample points to be right endpoints—in which case . * i i x x = AREAS BETWEEN CURVES
Background image of page 7

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

View Full Document Right Arrow Icon
The Riemann sum is therefore an approximation to what we intuitively think of as the area of S . This approximation appears to become better and better as n → ∞. [ ] 1 ( *) ( *) n i i i f x g x x = - AREAS BETWEEN CURVES
Background image of page 8
Thus, we define the area A of the region S as the limiting value of the sum of the areas of these approximating rectangles. The limit here is the definite integral of f - g . [ ] 1 lim ( *) ( *) n i i n i A f x g x x →∞ = = - AREAS BETWEEN CURVES Definition 1
Background image of page 9

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

View Full Document Right Arrow Icon
Thus, we have the following formula for area: The area A of the region bounded by the curves y = f ( x ), y = g ( x ), and the lines x = a, x = b , where f and g are continuous and for all x in [ a , b ], is: ( ) ( ) f x g x ( 29 ( 29 b a A f x g x dx = - AREAS BETWEEN CURVES Definition 2
Background image of page 10
Notice that, in the special case where g ( x ) = 0, S is the region under the graph of f and our general definition of area reduces to Definition 2 in Section 5.1 AREAS BETWEEN CURVES
Background image of page 11

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

View Full Document Right Arrow Icon
f and g are positive, you can see from the figure why Definition 2 is true: [ ] [ ] [ ] area under ( ) area under ( ) ( ) ( ) ( ) ( ) b
Background image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 53

Chap6_Sec1 - 6 APPLICATIONS OF INTEGRATION APPLICATIONS...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online