6.3 Point of View: Integrating along the y-axis

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1 math appliation: area between urves 7 EXAMPLE 66 Find the area of the region in the first quadrant enlosed by the graphs of y =, y = ln x, and the x- and y-axes SOLUTION It is easy to sketh the region See Figure 6 The urve y = ln x intersets the x-axis at x = and the line y = at x = e Notie that the bottom urve of the region swithes from x-axis to y = ln x at x = The region is divided into two y = ln x subregions (one is a square!) and the graph gives the relative positions of the urves Sine both the funtions are ontinuous Theorem 6 applies e Z Z e Area = dx + ln xdx We an rewrite the integral in a more onvenient way Notie that the area the we are trying to find is really just the retangle of height minus the area under y = ln x on the interval [, e] (Yet another way of saying this is that we are splitting R e ln xdx into two integrals R e dx and R e ln xdxand then ombining the two integrals R dx + R e dx into one leaving R e ln xdx) We get Z e Z e Area = dx ln xdx= e +??? The problem is that we do not know an antiderivative for ln x So we need another way to attak the problem We desribe this below y = Figure 6: The region in the first quadrant enlosed by the graphs of y =, y = ln x, and the x- and y-axes There are two representative retangles beause the bottom urve hanges 6 Point of View: Integrating along the y-axis Reonsider Example 66 and hange our point of view Suppose that we drew our representative retangles horizontally instead of vertially as in Figure 6 The integration now takes plae along the y-axis on the interval [, ] Using inverse funtions, the funtion y = ln x is viewed as x = g(y) =e y Now the width of a representative retangle is Dy and the (horizontal) height of the ith suh retangle is given by g(y i ) As we saw earlier in the term with integration along the x-axis, sine g is ontinuous, the exat area of the region is given by Area = lim n n! Â i= g(y i )Dy = Z d g(y) dy g(y i )! Dy { x = e y (y = ln x) Figure 6: The region in the first quadrant enlosed by the graphs of y =, y = ln x, and the x- and y-axes There are two representative retangles beause the bottom urve hanges e In our partiular ase, the interval [, d] = [, ] along the y-axis The funtion g(y) =e y So the area of the region is in Figure 6 (or equivalently 6) is Area = Z d g(y) dy = Z e y dy = e y = e We an generalize the argument we just made and state the equivalent of Theorem 6 for finding areas between urves by integrating along the y-axis THEOREM 6 (Integration along the y-axis) If f (y) and g(y) are ontinuous on [, d] and g(y) apple f (y) for all y in [, d], then the area of the region bounded by the graphs of x = f (y) and x = g(y) and the horizontal lines y = and y = d is d Z d Area beween f and g = [ f (y) g(y)] dy g(y) x = f (y) Figure 6: The region bounded by the graphs of x = f (y) and x = g(y) and the horizontal lines y = and y = d

2 math appliation: area between urves 8 6 More Examples Here are a few more examples of area alulations, this time involving integrals along the y-axis EXAMPLE 67 Find the area of the region in the first quadrant enlosed by the graphs of x = y and x = y + SOLUTION The intersetions of the two urves are easily determined: y = y + ) y y = ) (y + )(y ) = ) y =, It is easy to sketh the region sine one urve is a parabola and the other a straight line See Figure 6 Sine both the funtions are ontinuous Theorem 6 applies x = y x = y + Z d Z Area beween f and g = [ f (y) g(y)] dy = [y + y ] dx = y + y y = = 9 Figure 6: The region in the first quadrant enlosed by the graphs of x = y and x = y + EXAMPLE 68 Find the area of the region enlosed by the graphs of y = artan x, the x-axis, and x = SOLUTION The region is given to us in a way that simply requires skething artan x The area is desribed by Z Area = artan xdx However, we don t yet know an antiderivative for the artangent funtion We ould develop that now (or look it up in a referene table), or we an swith that axis of integration Notie that y = artan x () x = tan y The old limits were x = and x =, so the new limits for y are artan = and artan = p Notie the funtion x = is the top urve and x = tan y is the bottom urve (reading from left to right) Sine both the funtions are ontinuous Theorem 6 applies Z p/ p Area = tan ydy= y ln se y p = ln p ( ln ) = p ln EXAMPLE 69 Find the area of the region enlosed by the three graphs y = x, y = 8 x, and y = (The region is enlosed by all three urves at the same time) SOLUTION Determine where the three urves meet: x = 8 x ) x = 8 ) x = x = ) x = (not, see Figure 65) 8 x = ) x = 8 Notie from Figure 65 that if we were to find the area by integrating along the x-axis, we would need to split the integral into two piees beause the top urve of the region hanges at x = We an avoid the two integrations and all of the orresponding evaluations by integrating along the y-axis We need to onvert the funtions to funtions of x in terms of y: y = x ) x = p y and y = 8 x ) x = 8 y p y = artan x (x = tan y) x = Figure 6: The region in the first quadrant enlosed by the graphs of y = artan x, the x-axis, and x = Integrate along the y-axis and use x = tan y y = x y = 8 x x = 8 Figure 65: The region enlosed by the three graphs y = x, y = 8 x, and y = Integrating along the y-axis uses only a single integral

3 math appliation: area between urves 9 Remember to hange the limits: At x =, y = and at x = or x = 8, y = Notie the funtion x = 8 y is the top urve and x = p y is the bottom urve (reading from left to right) Sine both the funtions are ontinuous Theorem 6 applies Z 8 Area = y p ydy= 8 ln y x / = 8 ln 6 ln = 8 ln YOU TRY IT 6 Redo Example 69 using integration along the x-axis Verify that you get the same answer Whih method seemed easier to you? YOU TRY IT 65 Set up the integrals that would be used to find the shaded areas bounded by the urves in the three regions below using integration along the y-axis You will need to use appropriate notation for inverse funtions, eg, x = f (y) 5 5 x = g(y) x = f (y) 5 x = g(y) x = f (y) Shaded x = f (y) 5 6 Shaded 5 6 Shaded 5 6 YOU TRY IT 66 Sketh the regions for eah of the following problems before finding the areas (a) Find the area enlosed by x = y + and x = y + 9 Integrate along the y-axis (Answer: 6) (b) Along the y-axis (more in the next problem) The area enlosed by y = x y = x (Answer: 8) and () Find the area in the first quadrant enlosed by the urves y = p x, y = x, the x-axis, and the y-axis by using definite integrals along the y-axis (Answer: /, if you get 9/, you have the wrong region) (d) Find the area of the wedge-shaped region below the urves y = p x, y = x, and above the x-axis Integrate along either axis: your hoie! (Note: Not the same as (b)); Answer: 7/6) SOLUTION We do part () The urves are easy to sketh; remember y = p x is the graph of y = p x shifted to the right unit To integrate along the y axis, solve for x in eah equation These urves interset when y = p x ) y = x ) x = y + y = x ) x = y y = y ) y + y =(y )(y + ) = ) y = (not ) Of ourse x = Z Area = y intersets the y-axis at So Z y + dy + ydy= y + y + y y = YOU TRY IT 67 Sketh the region (use your alulator?) and find the area under y = arsin x on the interval [, ] Hint: swith axes (Answer: p/ ) YOU TRY IT 68 (From a test in a previous year) Consider the region bounded by y = ln x, y =, and y = x shown below Find the area of this region (Answer: e 5) y = x y = ln x e y = x y = p x Figure 66: The region enlosed by y = p x, y = x, the x-axis and the y-axis

4 math appliation: area between urves YOU TRY IT 69 Find the area of the region in the first quadrant enlosed by y = 9 x, y = x p x +, and the y-axis Hint: The two urves meet at the point (, 6) YOU TRY IT 6 Find the region enlosed by the three urves y = x, y = x x + 8, and y = x You will need to find three intersetions (Answer: 8) YOU TRY IT 6 Extra Fun (a) (Easy) The region R in the first quadrant enlosed by y = x, the y-axis, and y = 9 is shown in the graph on the left below Find the area of R (b) A horizontal line y = k is drawn so that the region R is divided into two piees of equal area Find the value of k (See the graph on the right below) Hint: It might be easier to integrate along the y-axis now Answer: (5) / 9 9 k YOU TRY IT 6 Let R be the region enlosed by y = x, y = x+, and the y axis in the first quadrant Find its area Be areful to use the orret region: One edge is the y axis (Answer: ln() ) YOU TRY IT 6 Two ways (a) Find the area in the first quadrant enlosed by y = p x, the line y = 7 x, and the x-axis by integrating along the x-axis Draw the figure (Answer: ) (b) Do it instead by integrating along the y-axis () Whih method was easier for you? YOU TRY IT 6 Find the area of the region R enlosed by y = p x, y = p x, and the x-axis in the first quadrant by integrating along the y axis Be areful to use the orret region: One edge is the x axis (Answer: 8) YOU TRY IT 65 (Good Problem, Good Review) Find the area in the first quadrant bounded by y = x, y =, the tangent to y = x at x = and the x-axis Find the tangent line equation Draw the region Does it make sense to integrate along the y-axis? Why? (Answer: ) YOU TRY IT 66 (Extra Credit) Find the number k so that the horizontal line y = k divides the area enlosed by y = p x, y =, and the y axis into two equal piees Draw it first This is easier if you integrate along the y axis YOU TRY IT 67 (Real Extra Credit) There is a line y = mx through the origin that divides the area between the parabola y = x x and the x axis into two equal regions Find the slope of this line Draw it first The answer is not a simple number

5 Two Appliations to Eonomis 65 An Appliation of Area Between Curves: Lorenz Curves In the last few years, espeially during the presidential eletion, there was muh talk about "the %" meaning "the wealthiest % of the people in the ountry," and the rest of us, "we are the 99%" Suh labels were intended to highlight the inome and wealth inequality in the United States Consider the following from the New York Times The top perent of earners in a given year reeives just under a fifth of the ountry s pretax inome, about double their share years ago (from wwwnytimesom///5/business/the--perent-paint-a-more-nuaned-portrait-of-the-rih html The wealthiest perent took in about 6 perent of overall inome 8 perent of the money earned from salaries and wages, but 6 perent of the inome earned from self-employment They ontrolled nearly a third of the nation s finanial assets (investment holdings) and about 8 perent of nonfinanial assets (the value of property, ars, jewelry, et) (See measuring-the-top--by-wealth-not-inome/) When we make statements suh as x% of the population ontrols y% of the wealth in the ountry, we are atually plotting points on what eonomists all a Lorenz urve DEFINITION 65 The Lorenz Curve L(x) gives the proportion of the total inome earned by the lowest proportion x of the population It an also be used to show distribution of assets (total wealth, rather than inome) Eonomists onsider it to be a measure of soial inequality It was developed by Max O Lorenz in 95 for representing inequality of the wealth distribution EXAMPLE 65 L(5) = would mean that the poorest 5% of households earns % of the total inome L(9) =55 would mean that the poorest 9% earns 5% of the total inome Equivalently, the rihest % households earn 5% of the total inome Fousing on wealth rather than inome, if the top % households ontrol about a third of the nation s finanial assets as the New York Times indiates, then the bottom 99% ontrol about two-thirds of the nation s wealth This would be represented on the Lorenz urve by the point L(99) =67 Basi Properties of the Lorenz Curve about the Lorenz urve There are a ouple of simple observations The domain of Lorenz urve is [, ]; any perent is expressed as a deimal in this interval For the same reason, the range of Lorenz urve is [, ] So the graph of a Lorenz urve lies inside the unit square in the first quadrant

6 math appliation: area between urves L() = sine no money is earned by households L() = beause all of the inome is earned by the entire population L(x) is an inreasing funtion More of the total inome is earned by more of the households Extreme Cases Two extreme ases that help us understand the Lorenz urve Absolute Equality of Inome Everyone earns exatly the same amount of money In this situation L(x) =x, that is, x% of the people earn x% of the inome Absolute Inequality of Inome Nobody8earns any inome exept one person (who <, for apple x < earns it all) In this situation L(x) = :, for x = Let s think about this a bit The lowest paid x% of the population annot earn more than x% of the inome, therefore, L(x) apple x (If they did, the remaining ( x)% would earn less than ( x)% of the total inome and would be lower paid than the x%) This means that the Lorenz urve L(x) lies at or under the diagonal line y = x in the unit square A typial Lorenz urve is shown to the right in Figure 68 The information in a Lorenz urve an be summarized in a single measure alled the Gini index DEFINITION 65 Let A be the area between the line y = x representing perfet inome equality and the Lorenz urve y = L(x) (This is the shaded area in Figure 68) Let B denote the region under the Lorenz urve Then the Gini index is % Inome 8 6 A B 6 8 % Households Figure 67: A Lorenz urve L(x) ompared to the diagonal line whih represents perfet inome equality The region A represents the differene between perfet inome equality and atual inome distribution The larger A is, the more unequal the inome distribution is y = L(x) G = A A + B The area A + B = beause it is half of the unit square So G = A = A Using definite integrals to alulate the area of A, we find Z G = A = x L(x) dx YOU TRY IT 68 Using properties of the integral prove that Z G = L(x) dx Notie that when there is perfet inome equality, L(x) =x is the diagonal and we have A =, so G = A = When there is absolute inome inequality, then B = and A =, so G = G always falls in the range of to with values loser to representing more equally distributed inome Problems Several of these problems are taken almost word-for-word diretly from middleburyedu/mathc/lorenz%curvespdf

7 math appliation: area between urves A very simple funtion used to model a Lorenz urve is L(x) =x p, where p (a) Chek that any suh funtion L(x) =x p, where p is a valid Lorenz urve (That is, hek that L() =, L() = and L(x) apple x on [, ]) (b) Use a graphing alulator or omputer to examine the graphs of L(x) =x p on the interval [, ] for p =, 5,,, and Whih value of p gives the most equitable distribution of inome? The least? A ountry in Northern Europe has a Lorenz urve for household inomes given by the funtion L(x) = ex e for apple x apple (a) Show that this funtion is a valid andidate to be a Lorenz urve (That is, hek that L() =, L() = and L(x) apple x) (b) Determine L(5) and interpret what it means () What is the Gini oeffiient for this ountry? Give your interpretation and ommentary (d) The CIA wesbite reports the Gini index for the distribution of family inome in the United States to be 5 Roughly how does the inome inequality you omputed ompare to that in the US at the present time? Find the Gini index orresponding to the Lorenz urve f (x) =x Find the Gini index orresponding to the Lorenz urve f (x) = x + x 5 Prove that a Lorenz urve of the form L(x) =x p has a Gini index of G = p p+ 6 The CIA wesbite reports the Gini index for the distribution of family inome in the United States to be 5 (a) Determine the number p so that the Gini index is 5 if the Lorenz urve has the form of a power funtion f (x) =x p (b) Aording to this model, how muh of the family inome is earned by the top 5% of families? 6 8 % Inome 6 8 % Households A B y = L(x) Figure 68: The Lorenz urve for the United States based on data from RS8_pdf 7 One type of funtion often used to model Lorenz urves is L(x) =ax +( a)x p Suppose that a = and that the Gini index for the distribution of wealth in a ountry is known to be 9 6 (a) Find the value of p that fits this situation (b) Aording to this model, how muh of the wealth is owned by the wealthiest 5% of the population? 8 Two-lass soieties In theory, it ould happen that one portion of the total resoures is distributed equally among one lass, with the rest being shared equally by another lass Here are funtions that represent two different twolass soieties: L (x) = 8 < : x, for apple x apple x, for apple x apple L (x) = 8 < : x, for apple x apple 5x, for apple x apple Compute the Gini index for eah and deide whih is the more equitable soiety In eah ase, how muh of the total resoures are owned by the rihest half of the population?

8 math appliation: area between urves 66 An Appliation of Area Between Curves: Consumer Surplus The material up to Figure 69 is taken almost word-for-word diretly from http: //tutorunet/eonomis/revision-notes/as-markets-onsumer-surplushtml In this note we look at the importane of willingness to pay for different goods and servies When there is a differene between the prie that you atually pay in the market and the prie or value that you plae on the produt, then the onept of onsumer surplus is useful Defining onsumer surplus Consumer surplus is a measure of the welfare that people gain from the onsumption of goods and servies, or a measure of the benefits they derive from the exhange of goods DEFINITION 6 Consumer surplus is the differene between the total amount that onsumers are willing and able to pay for a good or servie (indiated by the demand urve) and the total amount that they atually do pay (ie, the atual market prie for the produt) The level of onsumer surplus is shown by the area below the demand urve and above the ruling market prie line as illustrated in Figure 69 Produer surplus is the differene between the total amount that produers of a good reeive and the minimum amount that they would be willing to aept for the good (lighter shading) P Prie Q Quantity Demand Equilibrium Prie Supply Consumer Surplus Produer Surplus Figure 69: Consumer surplus is the differene between the total amount that onsumers are willing and able to pay for a good or servie and the total amount that they atually do pay (darker shading) Produer surplus is the differene between the total amount that produers of a good reeive and the minimum amount that they would be willing to aept for the good (lighter shading) Consumer and produer surpluses are relatively easy to alulate if the supply and demand urves are straight lines However, in realisti models of the eonomy supply and demand generally do not behave in this way To determine either surplus for a produt, we first need to determine the equilibrium prie for that produt, that is, the prie for a good at whih the suppliers are willing to supply an amount of the good equal to the amount demanded by onsumers Typially, as in Figure 69, onsumers will demand more of a good only if the prie (vertial axis) is lowered So the demand urve has a negative slope On the other hand, produers will be willing to make and sell more of a good if the prie paid inreases So the supply urve generally has a positive slope The two urves meet at some quantity Q and some orresponding prie P There are examples of so-alled bakward-bending supply urves, where the supply urve inreases for awhile and then when a partiular prie is reahed suppliers atually are willing to produe less supply so the urve ontinues upward but bends bak to the left Extra redit if you an think of and justify an example where this might be true

9 math appliation: area between urves 5 EXAMPLE 6 Suppose that the demand urve for students wanting to attend Hobart and William Smith eah year is Demand Prie = 8 q+ where q is measured in thousands of students and demand prie is measured in thousands of dollars (Eg, no students are interested in attending if the tuition is 8 + = 9 thousand dollars, whereas thousand students are interested in attending if the tuition is 8 + = thousand dollars) The Colleges are willing to aept students aording to the formula for Supply Prie = 56q q+ Determine the equilibrium prie, make a quik sketh of the graphs, and then determine the onsumer surplus SOLUTION The equilibrium prie is where the demand prie equals the supply prie, 8 q + = 56q q + ) 56q + q = 8q + 8 ) q 7q 5 = Using the quadrati formula we find that the only positive root is q = 5 and the orresponding prie is p = 8 5+ = thousand dollars 9 Prie 5 Quantity Demand Equilibrium Prie Supply Consumer Surplus Produer Surplus Figure 6: The supply and demand urves for enrollment at HWS (in thousands of dollars and thousands of students) The onsumer surplus is the area below the demand urve and above the equilibrium prie p = So Consumer Surplus = Z 5 8 q + dq = 8 ln q + q 5 =(8 ln 5 ) (8 ln ) The onsumer surplus is approximately $5,967,89 (sine the units are thousands times thousands) YOU TRY IT 69 (Extra Credit) Find the produer surplus for the situation in Example 6 YOU TRY IT 6 (Extra Credit) Suppose that the Federal Government wants to inrease the number of students able to attend ollege and offers every suh student $, per year (in the form of a oupon payable to the the student s ollege) This means that students have another $, to spend per year, so they are willing to aept higher pries Their new Demand Prie = 8 q+ + (The + represents the extra $,) The Colleges supply prie remains the same (a) How many students now would attend the Colleges? You will need to use the quadrati formula Note: Round your answers for q to the nearest one-thousandth (and your final answer to the nearest student) (b) Find the new equilibrium prie (tuition) at HWS Did it go up $? () Find the new onsumer surplus Did it go up or down? (d) Find the new produer surplus Did it go up or down?