[br][b][u]Estimating The Area Under a Curve[/u][/b][br][br](You need a computer that has GeoGebra loaded)[br][b]1. [br][/b]Open up the files from above[br][b]a. [/b][b]Area by rectangles.ggb[/b][br][b]b. [/b][b]upper_lower_rectangles.ggb[/b][br][br][b]2. [br][/b]Using the file, [b]upper_lower_rectangles.ggb[/b], type in the box for the rule, [i]x[/i][sup]2[/sup] (type: x ^ 2)[br][b]a. [/b]Make the lower bound 0 (use thescroll bar up the top left).[br][b]b. [/b]Make the upper bound 4.[br][b]c. [/b]Make the number of divisions, n, 4 (this will make rectangles of width 1 unit).[br][b]d. [/b]Click on the Green checkbox (Show Upper Rectangles). Upper rectangles are made by using the upper (higher) [i]y [/i]value for each division.[br][b]f. [/b]Calculate the area of each rectangle (A = L x W). [br][b]g. [/b]What is the total area?[br][b]h. [/b]Is this greater or less than the actual area under the curve from 0 to 4?[br][b]i. [/b]Using another colour and the applet repeat for lower rectangles. (Lower rectangles are made by using the lower [i]y[/i] value for each division[br][b]j. [/b]The actual area will be between these two answers. Calculate it by using calculus.[br][list=1][list=a] [*]Now change the value of [b][i]n [/i][/b]and complete the table:[/*][/list][/list][table] [tr] [td][br] [b][i]n[/i][/b][br] [/td] [td][br] [b]Width[/b][br] [/td] [td][br] [b]Upper Rectangles[/b][br] [/td] [td][br] [b]Lower Rectangles[/b][br] [/td] [td][br] [b]Actual Area[/b][br] [/td] [/tr] [tr] [td][br] 4[br] [/td] [td][br] 1[br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 8[br] [/td] [td][br] [br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 20[br] [/td] [td][br] [br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 40[br] [/td] [td][br] [br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 100[br] [/td] [td][br] [br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr][/table][br][b]l. [/b]What do you notices as the number of divisions increases?[br][br][b]3. [/b]Repeat the process for the following functions:[br][br][b]a. [math]y=16-x^2[/math][br][/b][b]b. [math]y=x^3[/math] [/b] (use 3 divisions initially, then 6, 15, 30, 100)[br][br][br][table] [tr] [td][br] Rule[br][br] [/td] [td][br] Upper Rectangles[br][br] [/td] [td][br] Lower Rectangles[br][br] [/td] [/tr] [tr] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr][/table][br][b]In Summary:[/b][br]The area under a curve can be estimated by using _______ and _______ rectangles. As[br]the _______ of the rectangles ___________ the approximation is __________.[br][br][b]4. [br][/b]Using the file, [b]Areas by rectangles.ggb[/b] , type in the rule, [math]25-x^2[/math] (type: 25-x ^ 2)[br][br][b]a. [/b]Make the lower bound 0 and the upper bound 5 and divisions 5.[br][b]b. [/b]Complete:[br][br][table] [tr] [td][br] n[br][br] [/td] [td][br] width[br][br] [/td] [td][br] Upper[br][br] [/td] [td][br] Lower[br][br] [/td] [td][br] Left[br][br] [/td] [td][br] Right[br][br] [/td] [td][br] Actual[br][br] [/td] [/tr] [tr] [td][br] 5[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 10[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 20[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 50[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 100[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr][/table][br][br]Type in the rule, (type: 5x - x ^ 2)[br][br]a. Make the lower bound 0 and the upper bound 5 and divisions 5.[br]b. Complete:[br][table] [tr] [td][br] n[br][br] [/td] [td][br] width[br][br] [/td] [td][br] Upper[br][br] [/td] [td][br] Lower[br][br] [/td] [td][br] Left[br][br] [/td] [td][br] Right[br][br] [/td] [td][br] Actual[br][br] [/td] [/tr] [tr] [td][br] 5[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 10[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 20[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 50[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 100[br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr][/table][br][br][b]c. [/b]Are the Upper Rectangles always equal to the Right rectangles?[br][br][br][b]d. [/b]Explain the difference between Upper,Lower, Left and Right rectangles.[br][br][br][b]5. [/b]Using the file, [b]Areas by rectangles.ggb[/b] , type in the rule, (type: e ^ (0.5x) )[br][br][b]a. [/b]Make the lower bound 0 and the upperbound 5 and divisions 5.[br][br][b]b. [/b]Complete:[br][br][br][table] [tr] [td][br] n[br] [/td] [td][br] width[br] [/td] [td][br] Upper[br] [/td] [td][br] Lower[br] [/td] [td][br] Left[br] [/td] [td][br] Right[br] [/td] [td][br] Actual[br] [/td] [/tr] [tr] [td][br] 5[br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 10[br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 20[br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 50[br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr] [tr] [td][br] 100[br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [td][br] [br][br] [/td] [/tr][/table][br][b]c. [/b][b]What is the area enclosed by [math]y=e^{0.5x}[/math], the [img width=13,height=15]file:///C:/Users/AWA/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif[/img]-axis and the lines [img width=37,height=19]file:///C:/Users/AWA/AppData/Local/Temp/msohtmlclip1/01/clip_image034.gif[/img] and [img width=36,height=19]file:///C:/Users/AWA/AppData/Local/Temp/msohtmlclip1/01/clip_image036.gif[/img]?(width 1 unit)[/b][br][br] [br]i. Counting Squares; ( _____ sq. unit)[br][br] [br]ii. Left – endpoint rectangles; ( _____ sq. unit)[br][br] [br]iii. Right – endpoint rectangles;( _____ sq. unit)[br][br] [br]iv. Calculus ( _____ sq. unit).[br][br][br][br][br] [br][br][br]