1. Section 4-2-B More approximations of area using
Midpoint formula and Trapezoidal Rule

2. using the midpoints of rectangles
Approximating Area
using the midpoints of rectangles
Over Estimate: when concave down
Under Estimate: when concave up

3. 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:
Value of function between
f(x)0 and f(x)1. The leftmost
endpoint and the second x-value.
Value of function between
f(x)n-1 and f(x)n. The rightmost
endpoint and the second to
last x-value
Width of each rectangle
along the x-axis

4. Steps for using The Midpoint Formula
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. (height)
Fill in the Midpoint Formula

5. 10) Approximate the area under the curve from x = 0 to x = 6 with 6 rectangles using the midpoints.

6. 11) Approximate the area under the curve from x = 1 to x = 5 with 4 rectangles using the midpoints.

7. Trapezoidal Approximation for Area
Intermediate sides are shared by two trapezoids

8. Trapezoidal Rule: Let n be the number of trapezoids used on the interval [a,b]. Then the area approximated is given by:
Every intermediate value
is used twice so multiply
by 2
Endpoints only
used once
Width along
x-axis

9. Trapezoidal Approximations
Under Estimate: when concave down
Over Estimate: when concave up

10. 12) Approximate the area under the
curve from x = 0 to x = 4
with 4 trapezoids.

11. 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 10
When given the information in tabular form, verify the trapezoids have same width before using the Trapezoidal Rule Formula.

14. 16) A metal wire of length 8 centimeters is heated at one end
16) A metal wire of length 8 centimeters is heated at one end. The table gives selected values of the temperature T(x) in degrees Celsius of the wire x cm from the heated end. The function T(x) is decreasing and twice differentiable. Estimate the average temperature of the wire using a trapezoidal sum with four subintervals indicated by the data in the table.

15. Homework
Worksheet: Area Approximations
wks 4-2