Review Problem – Riemann Sums

Use a right Riemann Sum with 3 subintervals to approximate the definite integral:

x 2dx0

3

Applications of the Definite Integral

Mr. Reed

AP Calculus AB

Finding Areas Bounded by Curves

To get the physical area bounded by 2 curves:1. Graph curves & find intersection

points – limits of integration2. Identify “top” curve & “bottom”

curve OR “right-most” curve & “left-most” curve

3. Draw a representative rectangle4. Set up integrand: Top – Bottom Right – Left

Finding Intersection Points

Set equations equal to each other and solve algebraically

Graph both equations and numerically find intersection points

Example #1

Find the area of the region between y = sec2x and y = sinx from x = 0 to x = pi/4.

Example #2

Find the area that is bounded between the horizontal line y = 1 and the curve y = cos2x between x = 0 and x = pi.

Example #3

From Text – p.240 - #16

Example #4

Find the area of the region R in the first quadrant that is bounded above by y = sqrt(x) and below by the x-axis and the line y = x – 2.

Summarize the process

AP MC Area Problem

#12 from College Board Course Description

Homework

P.236-240: Q1-Q10, 13-25(odd)

Authentic Applications for the Definite Integral

Example #2 – p.237

Definite Integral Applied to Volume

2 general types of problems:1.Volume by revolution2.Volumes by base

Volume by Revolution – Disk Method

The region under the graph of y = sqrt(x) from x = 0 to x = 2 is rotated about the x-axis to form a solid. Find its volume.

Volume by Revolution – Disk Method

Homework #1 – Disk Method about x and y axis

P.246-247: Q1-Q10,1,3,5

Volume by Revolution – About another axis

The region bounded by y = 2 – x^2 and y = 1 is rotated about the line y = 1. Find the volume of the resulting solid.

Volume by Revolution – Washer Method

Find the volume of the solid formed by revolving the region bounded by the graphs of f(x) = sqrt(x) and g(x) = 0.5x about the x-axis.

Homework #2 – Washer Method & Different axis

P.247 – 249: 7,9,11,14

Volume with known base

The base of a solid is given by x^2 + y^2 = 4. Each slice of the solid perpendicular to the x-axis is a square. Find the volume of the solid.

Homework #3 – Different axis & known base

P.249: 15,16,18,19