Warm Up – Calculator Active1) A particle moves along the x-axis so that at any

time t > 0, its velocity is given by v(t) = 3 + 4.1cos

(0.9t). What is the acceleration of the particle at time t =

4?2) 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.

tv t sin t e

Approximating the area under a curve

…or at least the area between the curve and the x-axis

Riemann SumsRiemann SumsAnd And

Trapezoidal RuleTrapezoidal Rule

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

Riemann Sums use rectangles to approximate

Right sum – using 3 rectangles(in this case also the upper sum)

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

Midpoint sum – 3 equal subintervals

To be more accurate…Use more rectangles!

Try finding the sum with 6 rectangles.

Let f(x) = 0.5x3 – 2x2 + x + 5

a) 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.

Trapezoidal sum – 3 equal subintervals

Approximate the area under the curve y = 1- cos x on the interval using n = 4 equal subintervals and

3,2 2

1) Lower sums2) Upper sums3) Midpoint sums4) Trapezoidal Rule

4.252

5.823

5.194

5.038

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.

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)

0 1201 1304 2256 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.

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.

tYears since 1990

M Debt(in billions of dollars)

0 1201 1304 2256 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.

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.

tYears since 1990

m Debt(in billions of dollars)

0 1201 1304 2256 250

10 640

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.