Area & Volume. Chapter 6.1 & 6.2 September 25, y = 1! x 2. Back to Area:

Transcription

1 Bck to Are: Are & Volume Chpter 6. & 6. Septemer 5, 6 We cn clculte the re etween the x-xis nd continuous function f on the intervl [,] using the definite integrl:! f x = lim$ f x * i )%x n i= Where fx i* ) is the height of rectngle nd x is the width of tht rectngle. {-)/n n is the numer of rectngles)} n Rememer tht the re ove the xis is positive nd the re elow is negtive. Set up the integrl needed to find the re of the region ounded y: y = x! x! nd the x-xis. Set up the integrl needed to find the re of the region ounded y:, the x-xis on [,]. y =! x! x! x!! x!! x

2 Are ounded y two curves Superimposing the grphs, we look t the re ounded y the two functions: Suppose you hve curves, y = fx) nd y = gx) Are under f is:! f x Are under g is:! gx! f x! gx = f x)! gx) top - ottom)* x The re ounded y two functions cn e found: Find the re of the region etween the two functions: y = x! x + nd y =!x + 6 [-,] y =!x + 6 y = x! x +! top $ A = & '! ottom $ & dx!!!!!!!!!!!! ) ) Are? [!x + 6)! x! x + )]dx =!x + x + = 9

3 Find the re ounded y the curves: nd y = x! x! 5 y = x + Find the re ounded y the curves: nd y = x! x! 5 y = x + Solve for ounds: x + = x! x! 5 = x! 5x! 6 = x + )x! 6) x = 6!!!x = Sketch the grph: 6 [x + )! x! x! 5)]dx 6 top - ottom)* x! x + 5x + 6 = Find the re of the region determined y the curves: x = y! nd y = x! Find the re of the region determined y the curves: x = y! nd y = x! In terms of y: [-,] Points -,-) & 5,) Grph? Solve for y: x = y! y = ± x + 6 Need Integrls! One from - to - nd the other from - to 5. Are? [ x + 6!! x + 6)]dx! 5 + x + 6! x!

4 Horizontl Cut insted: In terms of y: [-,] In Generl: Verticl Cut: Horizontl Cut: Right Function? Left Function? x = y + x = y! Are? [y + )! y! )]dy = 8! A =! top $ & '! ottom $ & dx!!!!!!!!!!!!!!!!!!! ) A = d c! right $ & '! left $ & dy!!!!!!!!!!!!!!!!!!!c ) d Find the Are of the Region ounded y y = x nd y = x Find the Are of the Region ounded y y = cosx), y = sinx), x =! nd the y-xis [,] [,π/] nd [π/, π/] y = x y = cosx) y = x Are? x! x Are? cosx)! sinx) y = sinx)

5 Find the Are of the Region ounded y y = cosx), y = sinx), x =! nd the y-xis Find the re of the Region ounded y y = x +,!!y = x + 6,!!x =!!!nd!!x = 5 Are? cosx)! sinx) [,π/] nd [π/, π/] y = sinx) y = cosx) + sinx)! cosx) Intervl is from -, 5 Functions intersect t x = - nd x = Grph? Top function switches times! This clcultion requires integrls! Find the re of the Region ounded y y = x +,!!y = x + 6,!!x =!!!nd!!x = 5 Find T so the re etween y = x nd y = T is /. Are? [x + )! x + 6)]dx! + [x + 6)! x + )]dx 5 +![x + ) x + 6)]dx Are?! T! T, T $ % T y = T T! x y = x T Tking dvntge of Symmetry T! x T Are must equl /: T! x = Ans: T = 9 5

6 Volume & Definite Integrls Find the volume of the solid whose ottom fce is the circle x + y! nd every cross section perpendiculr to the x-xis is squre. We used definite integrls to find res y slicing the region nd dding up the res of the slices. [-,] We will use definite integrls to compute volume in similr wy, y slicing the solid nd dding up the volumes of the slices. For Exmple y =! x y =!! x Length?! x!!! x ) =! x Find the volume of the solid whose ottom fce is the circle x + y! nd every cross section perpendiculr to the x-xis is squre. We use this length to find the re of the squre. Find the volume of the solid whose ottom fce is the circle x + y! nd every cross section perpendiculr to the x-xis is squre. Wht does this shpe look like? Length? =! x Are?! x ) Volume? )! x! x Volume?! x 6

7 Volumes: We will e given oundry for the se of the shpe which will e used to find length. We will use tht length to find the re of figure generted from the slice. The dy or dx will e used to represent the thickness. The volumes from the slices will e dded together to get the totl volume of the figure. Find the volume of the solid whose ottom fce is the circle x + y! nd every cross section perpendiculr to the x-xis is circle with dimeter in the plne. Length? =! x Are?! x ) )! x Volume? Rdius:! x! x Using the hlf circle x + y! [,] s the se slices perpendiculr to the x-xis re isosceles right tringles. [,] Length? =! x Are? Volume?! x )! x )! x Visul? 7