[b][color=#0000ff]The lower sum equals the right sum and the upper sum equals the left sum.The total area of the inscribed rectangles is the lower sum, and the total area of the circumscribed rectangles is the upper sum .To compute the area under a curve, we make approximations by using rectangles inscribed in the curve and circumscribed on the curve .[/color][/b]

[list=1][*][i][color=#0000ff]Students will be able to find the upper sum and lower sum of polynomial.[/color][/i][/*][/list]

[list=1][*][color=#9900ff][i][b]Enter the cubic polynomial f(x) = -0.5x^3+2x^2 + 1 in input bar,[/b][/i][/color][/*][*][color=#9900ff][i][b]Create two point A and B on x-axis to determine restrict area of polynomial,[/b][/i][/color][/*][*][color=#9900ff][i][b]Create slider n=Slider(1,50,1) by using input bar,[/b][/i][/color][/*][*][color=#9900ff][i][b]create uppersum = UpperSum[f, x(A), x(B), n] by using input bar,[/b][/i][/color][/*][*][color=#9900ff][i][b]create lowersum = LowerSum[f, x(A), x(B), n] by using input bar,[/b][/i][/color][/*][*][color=#9900ff][i][b]To find difference ,Enter difference = upersum-lowersum in input bar,[/b][/i][/color][/*][*][color=#9900ff][i][b]Create F=Integral[f, x(A), x(B)] by using input bar.[/b][/i][/color][/*][/list]

GGB applet

[list=1][*][i][b][color=#00ff00]Enter the cubic polynomial f(x) = -0.5x^3+2x^2 + 1,[/color][/b][/i][/*][*][i][b][color=#00ff00]Create two points A and B on the x-axis. Hint: These points will determine the interval which restricts the area between the function and the x-axis,[/color][/b][/i][/*][*][i][b][color=#00ff00]Using Input box, create slider n =Slider(1,50,1),[/color][/b][/i][/*][*][i][b][color=#00ff00]Using Input box , create uppersum = UpperSum[f, x(A), x(B), n],[/color][/b][/i][/*][*][i][b][color=#00ff00]Using Input box , create lowersum = LowerSum[f, x(A), x(B), n],[/color][/b][/i][/*][*][i][b][color=#00ff00]Insert dynamic text UpperSum = and select uppersum from Objects,[/color][/b][/i][/*][*][i][b][color=#00ff00]Insert dynamic text LowerSum = and select lowersum from Objects,[/color][/b][/i][/*][*][i][b][color=#00ff00]Calculate the difference diff = uppersum – lowersum,[/color][/b][/i][/*][*][i][b][color=#00ff00]Insert dynamic text Difference = and select diff from Objects,[/color][/b][/i][/*][*][i][b][color=#00ff00]Using Input box , create F = Integral[f, x(A), x(B)],[/color][/b][/i][/*][*][i][b][color=#00ff00]Insert dynamic text Integral = and select F from Objects[/color][/b][/i][/*][/list]