Section 7.1 Area of a Region Between Two Curves

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1 Section 7.1 Are of Region Between Two Curves

2 White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? cm

3 White Bord Chllenge Find the re of the region ounded y the function elow nd the -is etween = 1 to = 6: Clcultor f f 1 d d 1.16

4 Are Between Two Curves The re of region tht is ounded ove y one curve, y = f(), nd elow y nother y = g(). The re is lwys POSITIVE.

5 White Bord Chllenge Find the re of the region etween y = sec nd y = sin from = 0 to = π/4: y Sutrcting the ottom re from the top, leves only the re in-etween. y sin sec 4 Are etween the curves sec d 4 TOP TOP sec 4 0 BOTTOM sin BOTTOM sin d d Clcultor In this emple, ll of the re ws ove the -is. Does the sme process work for negtive re?

6 Are Between Two Curves: Positive nd Negtive Are Find the re of the region etween the two curves from = to = : Sutrcting the negtive re switches it to dding positive version. f Between (Positive) Between (Negtive) Must e positive! g Are etween the curves f d TOP TOP BOTTOM g d BOTTOM f g d THE SAME! In this emple, one re ws positive nd one ws negtive. Does the sme process work if oth res re negtive?

7 f Are Between Two Curves: Negtive Are Only Find the re of the region etween the two curves from = to = : Outside (Counted Twice) Between (Negtive) Sutrcting the negtive re switches it to dding positive version AND cncels the outside re. g Are etween the curves f d TOP TOP BOTTOM g d BOTTOM f g d THE SAME! In this emple, oth res were negtive. Now we cn pply the three scenrios to ny two curves.

8 Are Between Two Curves: A Mi Find the re of the region etween the two curves from = to = : f Are etween the curves g NEG-NEG f d TOP TOP BOTTOM g d BOTTOM f g d POS-POS POS-NEG

9 Are Between Two Curves If f nd g re continuous functions on the intervl [,], nd if f() g() for ll in [,], then the re of the region ounded ove y y = f(), elow y y = g(), on the left y =, nd on the right y = is: A f g d TOP BOTTOM

10 Reminder: Riemnn Sums Recll tht the integrl is limit of Riemnn Sums: f k * * k k f g Are n lim 0 m k k 1 * * f g k k k f g d g

11 Emple 1 Find the re of the region etween the grphs of the functions f g 4 10, 4, 1 3 Sketch Grph Bse = d Find the Boundries/Intersections 1,3 Mke Generic Riemnn Rectngle(s) Height = f g 1 3 Integrte the Are of Ech Generic Rectngle d 16 3

12 Emple Find the re of the region enclosed y the prols y = nd y =. Sketch Grph Bse = d Mke Generic Riemnn Rectngle(s) Height = ( ) ( ) Find the Boundries/Intersections ,1 Integrte the Are of Ech Generic Rectngle 1 0 d 1 3

13 Emple 3 Find the re of the region ounded y the grphs y = 8/, y = 8, nd y =. Sketch Grph Height Bse = 8/ = d Bse Height = d Integrte the Are of Ech = 8- Generic Rectngle Find the Boundries/Intersections d d 8 Mke Generic Riemnn Rectngle(s) 6

14 Emple 4 Find the re of the region ounded y the curves y = sin, y = cos, = 0, nd = π/. Sketch Grph Bse = d Bse = d Height = Height = cos-sin sin-cos Mke Generic Riemnn Rectngle(s) Find the Boundries/Intersections Integrte the Are of Ech Generic Rectngle cos sin sin cos 4 d d 0 4 0, sin cos Wht other Integrls could e used? 4 cos sin d (Symmetricl) 0 0 cos sin d 4 (Keeps it Positive)

15 NOTE: There hve een AP prolems in the pst tht sk for n integrl without n solute vlue. So the first method is still importnt. Are Between Two Curves If f nd g re continuous functions on the intervl [,], then the re of the region ounded y y = f(), y = g(), on the left y =, nd on the right y = is: A f g d It does not mtter which function is greter.

16 Wrm-up : 1985 Section I No Clcultor NOW WE CAN DO! d

17 White Bord Chllenge Find the re enclosed y the line y = 1 nd the prol y = + 6. Clcultor y d y d 3 18 y 6

18 Emple 5 Find the re enclosed y the line y = 1 nd the prol y = + 6. Sketch Grph Sometimes Solve for y 1 y 6 y1 1 y Mke Generic Riemnn Rectngle(s) Find the Boundries/Intersections Bse = dy y1 y 3 Height=(y+1) (1/y 3) Integrte the Are of Ech Generic Rectngle 4 1 y 1 y 3 dy y y8 0 y 4 y y,4 3

19 White Bord Chllenge Using two methods (one with d nd one with dy), find the re etween the -is nd the two curves: y & y y or y d d y or y OR 0 y y dy 10 3

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