Calculator Lab 7 for the Ti-Nspire
Riemann Sums and Definite Integrals
Name ______Date ______
Calculator Functions
From Calculator Screen Menu 4: Calculus 2: Numerical Integral
From Graph Screen Menu 6: Analyze Graph 6:Integral
Document
Riemann_Sums
You have algebraically calculated Riemann sums using left-hand and right-hand endpoints of an interval. Another Riemann sum uses rectangles whose heights are obtained by taking midpoints of each interval. The Ti-Nspire also has built in functions to calculate an approximation of the integral. In this lab you will try all of these methods and compare results.
Riemann_Sums Document Use it for calculating columns 2 thru 5 of the table that follows.
Go to My Documents and select the document named Riemann-Sums
You will be investigating several equations and intervals. The graph page is split horizontally into 2 screens. To move from the top screen to the bottom and vice versa, use CTRL TAB.
Values in the upper screen that can be changed by dragging the points to the desired values:
· The intervals (a and b)
· The number of rectangles (n)
· The Riemann sum type (L for Left, M for Middle, R for Right, T for Trapezoidal)
Values in the lower screen that can be changed by double clicking on the item
· f1(x)
From Calculator Screen
Use it for calculating column 6 of the table that follows.
Menu 4: Calculus 2: Numerical Integral
Fill in the left endpoint of interval, right endpoint of interval, the equation of the function, and the variable to differentiate with respect to
From Graph Screen
Use it for calculating column 7 of the table that follows.
Enter the equation for the function on the entry line.
Menu 6: Analyze Graph 6:Integral
When prompted for lower bound type the lower bound enclosed in parenthesis. Move hand to the right of the lower bound and type the upper bound enclosed in parenthesis.
| Riemann Sums | Calculator
Function / Interval / Left Sum / Midpoint / Right Sum / Trapezoid / Calc Screen / Graph Screen2x / -2,1
Sin x / 0,
-2,2
0,3
xe / 0,1
x / 0,3
4 cos x / 0,
Comparing the different Riemann sums with the Ti-Nspire built in functions, which Riemann sum seems to give the better approximation to the definite integral?
______
Why? ______
______
What happens if you increase the number of rectangles? ______
Looking at your results. Is the midpoint sum an over or under approximation if the function is
· decreasing and concave down ______
· decreasing and concave up ______
· increasing and concave up ______
· increasing and concave down ______
Why? ______
______
Technology Stretch
Remember Mattie from Lab 5. She also likes speed boats. Below is a graph of the acceleration of Mattie’s last 2 hour boat trip. You should know that Mattie started at time t = 0 with position 0 and velocity 0. Sketch her velocity and position graphs and see if you can come up with their equations.
Velocity equation ______
Position equation ______