# Areas - 6.1 Areas Between Curves Suppose we want to find...

• Notes
• nizhes
• 6

This preview shows pages 1–6. Sign up to view the full content.

6.1 Areas Between Curves Suppose we want to find the area of the region S as shown in Figure 1. f g x y b a Figure 1 S

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

f g x y b a [ ] [ ] dx x g x f x x g x f A b a n i i i n ) ( ) ( ) ( ) ( lim 1 - = - = = ) ( ) ( i i x g x f - x i x Figure 2
When the graph of the area of the region is already determined, it is often convenient to use a typical approximating rectangle (strip) to write the integration formula for the area between two curves. In general, we use ( 29 ( 29 [ ] - = 2 1 curve bottom curve top x x dx A Vertical Strips (of width ) implies integration with respect to x x Horizontal Strips (of width ) implies integration with respect to y y in variable x ( 29 ( 29 [ ] - = 2 1 curve left curve right y y dy A in variable y

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

Examples Sketch the region bounded by the given curves and find the exact area of the region. Example One: Example Two: 1 , 3 2 = - - = y x x y 2 / , 0 , 2 sin , sin π = = = = x x x y x y
Solutions: Example One Find the points of intersection of the graphs: 1 , 3 2 = - - = y x x y ( 29 ( 29 2 or 1 2 1 0 2 0 1 3 2 2 - = = + - = - + = + = - y y y y y y y y (2,1) (-1, -2) x y 1 + = y x 2 3 y x - = The area is: ( 29 ( 29 [ ] ( 29 2 9 4 2 3 8 2 2 1 3 1 2 2 3 2 1 3 1 2 1 2 2 3 2 1 2 2 = - - - + - - =

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

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

Unformatted text preview: +--= +--= +--=---∫ ∫ y y y dy y y dy y y A y ∆ Solutions: Example Two Find the points of intersection of the two curves: x y x y 2 sin , sin = = 2 / ; 3 / or , 2 1 cos or , sin ) 1 cos 2 ( sin sin cos sin 2 cos sin 2 sin 2 sin sin π ≤ ≤ = = = = ⇒-=-= = = x x x x x x x x x x x x x x x 2 / 3 / 1 ( 29 2 / 3 , 3 / x y x y 2 sin = x y sin = The total area is: 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 cos 2 1 cos cos 2 cos 2 1 ) 2 sin (sin ) sin 2 (sin 2 / 3 / 3 / 3 / 2 / 3 / = -+-- --+ +-- + --= +-+ +-=-+-= ∫ ∫ x x x x dx x x dx x x A x ∆ x ∆...
View Full 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