Section 5.3 : Finding Area Between Curves

Transcription

1 MATH 9 Section 5. : Finding Are Between Curves Importnt: In this section we will lern just how to set up the integrls to find re etween curves. The finl nswer for ech emple in this hndout is given for informtion onl. You will e required to solve sme prolems with finl nswers when we cover section 7.. A) Are Between Curves in [, ] f() f() g() g() Totl Are etween f() nd g() in [, ] f ( ) d - [ f ( ) g( )] d Are [ Top Function Bottom Function ] d g ( ) d Emple : Find the re of the region tht is enclosed, +, - nd. + + ) Grph the functions, nd +. Shde in the region in the intervl [-, ] + ) The function + is on the top in the entire intervl of [-, ] ) Are: A [ Top - Bottom ] d [( + ) ( + )] d - ( + + ) d - + 5

2 B) Are Between Curve nd the -Ais in [, ] (-is is sme s ) f() f() Top function is f(), Bottom is Are [ Top Function Bottom Function ] d [ f ( ) ] d f ( ) d Top function is, Bottom is f() Are [ Top Function Bottom Function ] d [ f ( )] d f ( ) d or Are f ( ) d Emple : Find the re of the region tht is enclosed the -is, the grph + 8 over the intervl [-, ]. ) Grph the functions + 8, nd (the -is). Shde in the region in the intervl [-, ] ) The function is on the top in the entire intervl of [-, ] - ) Are: A [ Top - Bottom ] d [( ) ( + 8)] d ( + + 8) d.67

3 C) Are Between Intersecting Curves When [, ] is not given, then we must determine the pproprite vlues from the grph. The re ounded etween two functions f () nd g() is from the intersection point on the left to the intersection point on the right. To find these points of intersection, we solve the eqution: f ( ) g( ) for. Emple : Find the re of the region tht is enclosed the grphs of the functions: f ( ) nd g ( ) ) Grph oth functions ) Find the points of intersection of the two curves equting the two equtions : nd : ( + )( ) Then - or ) Are: A [ Top - Bottom ] d [( ) ( )] d ( + + ) d Emple : Find the re of the region tht is enclosed + nd the - is. ) Grph the functions +, nd (the -is) ) Find the points of intersection equting the two equtions + nd : + ( )( ) Then or + ) Are: A [ Top - Bottom ] d [( ) ( + )] d ( + ) d.

4 More Emples: Emple 5: Find the re of the region tht is enclosed the grphs of the functions: nd ) Grph oth functions ) Find the points of intersection of the two curves equting the two equtions : nd Squre oth sides: ( ) Then or ) Are: A [ Top - Bottom ] d [ ] d. Emple 6: Find the re of the region tht is enclosed the grphs of the functions: + nd. ) Grph oth functions ) Find the points of intersection of the two curves equting the two equtions : + 8 ( ) ( )( + ) Then - or ) Are: A [ Top - Bottom ] d [( + ) ( )] d ( + 8) d

5 D) Are Between Curves With Multiple Points of Intersection (Crossing Curves) If neither grph f () or g() lies ove the other over the whole intervl, then we rek the re into two pieces. One on either side of the point t which the grphs cross nd then compute ech re seprtel. To do this, we need to know ectl where tht crossing point is solving the eqution: f ( ) g( ) for. Emple 7: Find the re of the region tht is enclosed the -is, over the intervl [-, ]. ) Grph the functions, nd (the - is). Shde in the region in the intervl [-, ] ) Find the points of intersection equting the two equtions nd : ( )( + ) The grph crosses the -is t: -, ) The totl re is the sum of the two res: [-, ] nd [, ] A [ Top - Bottom ] d + [ Top - Bottom ] d [( ) ( )] d + [( ) ()] d ( + ) d + ( ) d Note : If ou write the re s A ( ) d, then the nswer will e Note : You should get the correct nswer for the Totl Are s long s ech integrl produces positive re. This is one w to check our work. Emple 8: Find the re of the region tht is enclosed the -is,, - nd. ) Grph nd shde the region etween - nd ) Find the points of intersection: or ( )( + ) The grph crosses the -is t - nd - ) A ( ) d + ( ) d

6 Emple 9: Find the re of the region tht is enclosed the grphs of the functions:,, over the intervl [-, ]. ) Grph the functions, nd. ) Find the points of intersection of the two curves equting the two equtions : ( )( + ) The two grphs cross when - nd. ) The totl re is the sum of the two res: [-, ] nd [, ] A [( ) ( )] d + [( ) ( )] d ( + ) d + ( ) d Emple : Find the re of the region tht is enclosed the grphs of the functions:, ) Grph the functions, nd. ) Find the points of intersection of the two curves equting the two equtions : ( ) ( )( + ) The two cross t:, -or ) The totl re is the sum of the two res: [-, ] nd [, ] A [ Top - Bottom ] d + [( ) ( )] d + [( ) ( )] d [ Top - Bottom ] d (notice tht the nswer will e zero if one re is negtive) - 6