AP Calc: Chapter 5 The beginning of integral fun… Section 5.1 – Warm Up Complete the Quick Review on page 1 of your packets 

Section 5.1 - Estimating with Finite Sums Objective: Estimating distance traveled, using the Rectangular Approximation Method (RAM), and Finding the volume of a sphere.

Distance Traveled Explore the problem: A train moves along a track at a constant rate of 75 miles per hour from 7:00 am to 9:00 am. What is the total distance traveled by the train? Recall: distance = rate * time Graph: What shape is the graph? How do you find the area of such a shape?

Rectangular Approximation Method (RAM) NOTE: RAM is the same thing as finding Riemann Sums like we did last year (and our Michigan Maps!). We can distinguish between the 3 types (left endpoint, right endpoint, and midpoint) by the following abbreviations: LRAM (left), MRAM (midpoint), RRAM (right).

Example 1: A particle starts at x = 0 and moves along the x-axis with velocity v(t) = t2 for time . Where is the particle at t = 3? Methods to approximate?

Example 2: The graph of y = x2sin x on the interval [0, 3] is crazy. Lets Graph it and then estimate the area under the curve from x = 0 to x = 3.

Generalizations For an increasing function LRAM MRAM RRAM For a decreasing function

5.5 trapezoidal rule

Section 5.2 Definite Integrals

Anatomy of an Integral Function Stop Integral Change in x Start

Example 1: Express the area of the shaded region below with an integral.

Definite Integrals and Area Example 2: Evaluate by drawing a picture

: Example 3 . Find the exact area of

Example 4 Evaluate the integral .

. Ex. 5: Use area to find

For any integrable function = (area above the x-axis) – (area below the x-axis)

Example 6: Evaluate the following integrals without a calculator given that 1. 2. 3. 4. 5. 6. 7. 8.

Integrals on a calculator!!! Evaluate the following integrals numerically on your calculators: a) b) c) Fn int (function,x,upper,lower)

Section 5.1-5.2, 5.5 Refresher