1. Warm Up – Calculator Active
1) A particle moves along the x-axis so that at any time
t > 0, its velocity is given by v(t) = cos (0.9t).
What is the acceleration of the particle at time t = 4?
A particle moves along the x-axis so that at any time t, 0 < t < 5, its velocity is given by
When t = 0, the particle is at the origin. Write an expression for the position function, x(t), of the particle at any time t.

2. Riemann Sums And Trapezoidal Rule
Approximating the area under a curve …or at least the area between the curve and the x-axis
Riemann Sums
And
Trapezoidal Rule

3. The area under y = x2 + 1 on the interval [0,3]

4. Riemann Sums use rectangles to approximate
Right sum – using 3 rectangles
(in this case also the upper sum)

5. Left sum – using 3 equal subintervals (in this case also the lower sum)

6. Midpoint sum – 3 equal subintervals

7. Try finding the sum with 6 rectangles.
To be more accurate…
Use more rectangles!
Try finding the sum with 6 rectangles.

8. Let f(x) = 0.5x3 – 2x2 + x + 5
Sketch the graph to illustrate the area approximation, under f(x) on the interval [-1,5] ,found by using 6 equal subintervals and an Upper Riemann Sum.
b) Sketch the graph to illustrate the area approximation, under f(x) on the interval [-1,5] ,found by using 3 equal subintervals and a Right Riemann Sum.

9. Trapezoidal sum – 3 equal subintervals

10. Approximate the area under the curve y = 1- cos x on the interval using n = 4 equal subintervals and
1) Lower sums 2) Upper sums 3) Midpoint sums 4) Trapezoidal Rule
4.252
5.823
5.194
5.038

11. Debt(in billions of dollars)
Fannie Mae is a federal institution that lends money for home mortgages. The outstanding debt over a 10 year period is indicated in the table. t
Years since 1990 M
Debt(in billions of dollars)
120 1
130 4
225 6
250 10
640
Use the trapezoidal rule with the four subintervals indicated by the data in the table to approximate the total debt over the ten-year period.

12. Debt(in billions of dollars)
Fannie Mae is a federal institution that lends money for home mortgages. The outstanding debt over a 10 year period is indicated in the table. t
Years since 1990 M
Debt(in billions of dollars)
120 1
130 4
225 6
250 10
640
Use a upper Reimann sum with the four subintervals indicated by the data in the table to approximate the total debt over the ten-year period.

13. Debt(in billions of dollars)
Fannie Mae is a federal institution that lends money for home mortgages. The outstanding debt over a 10 year period is indicated in the table. t
Years since 1990 m
Debt(in billions of dollars)
120 1
130 4
225 6
250 10
640
Use a lower Reimann sum with the four subintervals indicated by the data in the table to approximate the total debt over the ten-year period.

14. Use a midpoint Riemann sum with four subintervals of equal length and values from the table to
approximate the area under the curve given by the data in the table.