Area and The Distance

The Bicycle Problem 1

• An aerobics instructor is designing a 1-hour workout for the stationary bicycle. For best results, the speed of the wheels should stay between 10 mph (miles per hour) and 15 mph. You will see five different workouts she is considering in this applet. For instance, in example ride 1 she will spend the first 30 minutes of her ride at 10 mph, and the last 30 minutes of the ride at 15 mph.

• The big question: How far has she traveled? Ok, ok, yes, the real answer is 0 miles since this is a stationary bicycle! But what is a good estimate for "virtual distance" traveled?

The Bicycle Problem 2

• An aerobics instructor has decided that the best 1-hour stationary bicycle workout occurs when the speed of the bicycle follows the gray curve shown in this applet. If the speed of the bicycle follows the curve, what will be the total (virtual) distance traveled? We will estimate the distance.

• The big question: How far has she traveled? Ok, ok, yes, the real answer is 0 miles since this is a stationary bicycle! But what is a good estimate for "virtual distance" traveled?

• Start by finding an overall overestimate/underestimate

More Distance Problems

An object is thrown upwards from the ground. The velocity of the object after t minutes is given by the expression

More Distance Problems

See its graph in the applet • When was the object going up? Down? • Estimate the distance traveled by the object when

it goes up/Down – Record data every second– Record data every half a second

• Estimate the net distance traveled by the object? • Estimate the total distance traveled by the object?

Expressions to Estimate Distance(Summation Notation)

• Interval [0,9]– Length of subintervals – Expression to estimate distance (net, total)– Summation notation– Use calculator to evaluate summation