Points, Regions, Distance and Midpoints - College Algebra Section
SECTION 4-2-B More area approximations. Approximating Area using the midpoints of rectangles.
-
Upload
bryce-leonard -
Category
Documents
-
view
221 -
download
0
description
Transcript of SECTION 4-2-B More area approximations. Approximating Area using the midpoints of rectangles.
SECTION 4-2-BMore area approximations
Approximating Area using the midpoints of rectangles
Midpoint Formula Let n be the number of rectangles used on the interval [a,b]. Then the area approximated using the midpoint is given by:
2
1225
23
21 ... nyyyy
nabA
Value of function betweeny0 and y1. The leftmostendpoint and the second x-value.
Value of function betweenyn-1 and yn. The rightmostendpoint and the second tolast x-value
Width of each rectangle along the x-axis
• Midpoint Approximations• Over Estimate: when concave down• Under Estimate: when concave up
• Graph the function on the interval
• Determine width of each rectangle and mark on the graph
• Find the midpoint between each mark and use it to find the function value
• Fill in the Midpoint Formula
Steps for using midpoint formula
10) Approximate the area under the curve from x = 0 to x = 6 with 6 rectangles using the midpoints.
xy
11) Approximate the area under the curve from x = 1 to x = 4 with 4 rectangles using the midpoints.
23 xy
Trapezoidal Rule: Let n be the number of trapezoids used on the interval [a,b]. Then the area approximated is given by:
nn
nn
yyyyynab
yyyyyyyynabA
1210
1322110
2...2221
(...)()()(21
21trap 21 bbhA
Width alongx-axis
Endpoints onlyused onceEvery intermediate value
is used twice so multiplyby 2
Trapezoidal Approximations• Under Estimate: when concave down• Over Estimate: when concave up
Intermediate sides used for two trapezoids
12) Approximate the area under the curve from x = 0 to x = 4with 4 trapezoids.
220 xxy
13) If g(x) is a continuous function, find the area from x = 1 to x = 8 with four trapezoids given the information below.
x 1 2 3 6 8
g(x) 4 12 8 6 10
When given the information in tabular form, verify the trapezoids have same width before using the Trapezoidal Rule Formula.
Simpson’s Rule: Let n be the number of subintervals (must be even) used on the interval [a,b]. Then the area approximated is given by:
nn yyyyyynabA
13210 4...4243
width alongx-axis
Endpoint onlyused onceEvery intermediate value
alternates (+4) then (+2)
Simpson’s Rule Approximations• Under Estimate: when concave down• Over Estimate: when concave up
14) Approximate the area under the curve from x = 0 to x = π with n = 4 using Simpson’s rule
xy sin
15) Which method will overestimate and which will underestimate the area under the curve
on the given interval 2,0on sin xy
)cos(' xy Increasing and Decreasing
)sin('' xy Concave up and Concave Down
Homework
Worksheet: Area Approximations wks 4-2