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