Clicker Question 1 Suppose the acceleration (in feet/sec/sec)

of a rocket which starts from rest is a (t ) = 36t 2. How far does it travel during the first 5 seconds? A. 360 feet B. 1250 feet C. 1875 feet D. 1500 feet/sec E. 1525 feet

Clicker Question 2 A car travelling at a speed of 90 feet/sec

now decelerates at a constant rate of 30 ft/sec/sec. How far does it travel (from the time the deceleration begins) before it stops? A. 90 feet B. 120 feet C. 135 feet D. 196 feet E. 270 feet

Integration (3/28/12)

Question: How can we “add up” all the values of a function f (x ) on some interval [a, b ]?

This is called “integrating f (x ) on [a, b ]”

Doesn’t seem to make sense since most functions have infinitely many values on an interval. (Which ones don’t?)

Interpreting Integration as Area under the Curve

If we have a graph of f (x ), we can interpret “adding up all values” as finding the area under the graph on the interval [a, b ].

Just as slope of a curve is a graphical interpretation of the derivative, area under the curve is a graphical interpretation of the integral.

Finding Areas

Question: How can we compute the area under a given function on a given interval?

Answer: Not at all obvious!! An easy case: If f is linear. Example: What is the area under

f (x ) = x + 4 on [0, 3]?

But what if f is not linear?

What is the area under f (x ) = x 2 + 4 on [0, 3]?????

Let’s estimate it by using a single trapezoid.

It turns out the exact answer 21 sq. units. (We don’t know how to do this yet.)

Let’s estimate it using 3 trapezoids!

Assignment for Wednesday

Work on Hand-in #3 (due Thursday 4:45) Do the following 3 exercises on areas:

1. Find the exact area under f (x ) = 6 – x on the interval [0, 4].

2. Estimate (to two decimal places) the area under f (x ) = x on [0, 4] usinga. 1 trapezoid b. 2 trapezoids c. 4 trapezoids(Note: Exact answer is 5 1/3)

3. Estimate (to two decimal places) the area under f (x ) = sin(x ) on [0, ] usinga. 2 trapezoids b. 4 trapezoids (Exact is 2)