Let R be the region in the first quadrant bounded by the graph of ##f(x) =sqrt(1+e^-x), the line x=2 and the x-axis, how do you use the trapezoidal rule with n=4 to approx the area?

##int_0^2sqrt(1+e^-x)dx ~~ A_T=2.391##

Given ##n=4## trapezoids to use, you can measure the width of each trapezoid needed to estimate the area. Divide the difference of your bounds by the number of trapezoids you have.

##Deltax = (2-0)/4## ##Deltax = 1/2##

The area in question ranges from## x = 0## (the y axis) to the line ##x = 2## and is bounded by the ##x## axis and the function. Starting at ##x = 0##, create a new trapezoid base by increments of ##Deltax##, which is ##1/2##. Your can draw a picture based on that information.

All you have to do now is find the area of each separate trapezoid drawn and total them. ##A_T =[(b_1 + b_2)/2] * h##

The trapezoids are sideways if you ever find yourself confused:

##A~~ [f(0) + f(1/2)]/2*(1/2) + [f(1/2) + f(1)]/2*(1/2) + [f(1) + f(3/2)]/2*(1/2) + [f(3/2) + f(2)]/2*(1/2)##

##A ~~ (1/4)[ f(0) + 2*f(1/2) + 2*f(1) + 2*f(3/2) + f(2)]##

##A~~2.391##