5.1 Areas and Distance

Area and Distance are fundamental to the concept of integral calculus; as slope of tangent line and rate of change are to the concept of differential calculus.

Limits and Area

The area bounded by the continuous function f, x-axis, x = a and x = b.

Approximate the area under a curve by summing rectangles and then taking the limit as the number of rectangles increases.

Transparency –

First find L₃ R₃, M₃.

Then find L₆, R₆, M₆. Which appears to be the best approximation?

Estimate the area under the graph of from x = 0 to x = 5 using five approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?

Repeat using left endpoint.

Generalize the area under from 0 to 5 - using n subintervals, and the value of f at the right endpoints.

The width of the rectangles

The height of the rectangle at the right endpoint

Where

Definition 2

The area A of region S under the graph of the continuous function f is the limit of the sum of approximating rectangles

Similar for left endpoint, midpoint, … any point on the interval,

Using summation notation:

#18 Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.

#20 Determine a region whose area is equal to the given limit. Do not evaluate the limit

Distance Traveled

Speedometer reading for a motor cycle at 12-second intervals are given in the table.

a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals.

b) Give another estimate using the velocities at the end of the time periods.

c) Are your estimates in part a) and b) upper and lower estimates? Explain

t(s)	0	12	24	36	48	60
v(ft/sec)	30	28	25	22	24	27

Assignment: 5.1, pg 298; 1-5 odd, 13-21 odd