AP CALCULUS - AB LECTURE NOTES MS. RUSSELL Sectio Number: 4. Topics: Area -Sigma Notatio Part: of Sigma Notatio Upper boud Recall ai = a+ a + a3 + L + a idex i= Lower boud Example : Evaluate each summatio. 6 a. i b. j i= j= 3 Summatio Properties kai = k ai Property : i= i= Property : ai + bi = ai + bi Property 3: = i= i= i= i= i= i= a b a b i i i i Summatio Formulas.. 3. 4. Example : Use sigma otatio to rewrite each of the followig. 5 a. + + 5 + + 5 + 3 +L+ 5 +5 b. + 4 + 6 +L+
Example 3: Use the properties of summatio to evaluate the sum. Use the summatio capabilities of a graphig utility to verify your result. 0 a. (i ) i= 0 b. i(i +) i= Example 4: Evaluate the followig summatio. (Usig summatio formulas) i + i= for = 0, 00, 000, 0000 The sum appears to approach a limit as. Recall limits at ifiity which applies to a variable x, where x. May of the same results hold true for limits ivolvig variable, where + Example 5: Fid limit of +3 as.
Idex Shift It may be ecessary or coveiet to start at a differet lower boud. The process of chagig the lower boud is called a idex shift. Let s say we have +5 ad we wat to start our idex at = 0 = Defie a ew idex, i = The, whe = i = 0 Ad whe = i = = Rewrite i terms of ew idex i ad the reset the idex letter = +5 = i=0 (i +)+5 = i+ i=0 i +7 = +7 i+ + =0 There is a shorthad method which is easier to remember but still requires the same process. Decreasig/Icreasig iitial value of idex by a set amout o All other s icrease/decrease by that set amout Example 6: Idex Shifts a) Write ar as a summatio that starts at = 0 = b) Write as a summatio that starts at = 3 = 3 +
AP CALCULUS - AB LECTURE NOTES MS. RUSSELL Sectio Number: Topics: Area Part: of -Right/Left Side Edpoits 4. -Upper ad Lower Sums -Limit Process Suppose we wated to compute the area of a circle much like the aciet Greeks wated to do a few thousad years ago. Yet, we do ot have a actual formula to fid this area. Usig the fact that we do have a formula for computig the area of a triagle, which of the two shapes below would give us a more accurate represetatio of the area of the circle ad why? Circle Circle Example : Cosider the fuctio f x ( ) x 5 = + o the iterval [0, ]. Fid two approximatios of the area lyig betwee f( x) = x + 5, the x-axis, x = 0, ad x = usig 5 sub-itervals that are rectagles. Draw i the represetative rectagles for each approximatio. Right Side Edpoits 5 y Left Side Edpoits 5 y 4 f x ( ) x 5 = + f( x) = x + 5 4 3 3 x x Upper ad Lower Sums I the previous example (Example ), we could say that the area we computed usig the Right-Side Edpoits, is the Lower Sum. This is because it produces a area that is less tha the actual area uder the curve. Likewise, the area we computed usig the Left-Side Edpoits, could be called the Upper Sum. This is because it produces a area that is greater tha the actual area uder the curve. DO NOT ASSUME RIGHT-SIDE ENDPOINTS ALWAYS PRODUCE A LOWER SUM!
It all depeds o whether f (x) is icreasig or decreasig. Whe usig sub-itervals, follow these guidelies for producig The width of each rectagle is always width = b a The legth (height) of each rectagle depeds o whether you are computig it usig right-side edpoits or left-side edpoits. Legth of rectagle (right-side edpoits) Legth of rectagle (left-side edpoits) legth = b a f a+ i legth = b a f a+ ( i ) Note: The left-side edpoit process is a bit tougher to do algebraically due to the (i ). Example : Fid the upper ad lower sums for the regio bouded by the graph of betwee x = 0 ad x =. f( x) = x ad the x-axis
The Limit Process Usig the limit process to fid area. Area = lim f (c i )Δx, x i c i x i where Δx = b a i= Example 3: Fid lim i= + i Example 4: Fidig Area by Limit Defiitio a) Fid the area of the regio bouded by the graph f (x)= x 3 ad the x-axis o [0,].
b) Fid the area of the regio bouded by the graph f (x)= 4 x ad the x-axis o [,]. c) Fid the area of the regio bouded by the graph f ( y)= y, the y-axis for 0 y. If you ca do these problems the you are defiitely