How do you use the midpoint rule to approximate the integral #-3x-8x^2dx# from the interval [-1,4] with #n=3#?

1 Answer
Mar 2, 2015

Divide the interval #[-1,4]# into #3# equal width strips.
Find the mid point of each strip and evaluate #f(x)# at that point.
Multiply #f#(strip midpoint#) xx #strip-width# to get strip area.
Add the strip areas to get approximation of integral.

If #[-1,4]# is divided into #3# equal strips then each strip will be
#5/3# wide.

The strip mid points will be at
#-0.1667, 1.5,# and # 3.1667#

#f(x) = - 3x - 8x^2#
#f(#midpoint of first strip#) = f(-0.1667) = 0.2778#
#f(#midpoint of second strip#) = f(1.5) = - 22.5#
#f(#midpoint of third strip#) = f(3.1667) = - 89.722#

Area of first strip #= 0.2778 xx 5/3 = 0.4630#
Area of second strip #= (- 22.5) xx 5/3 = -37.5#
Area of third strip #= (- 89.722) xx 5/3 = -149.537

Total area of strips
#0.4630 + (-37.5) + (-149.537)#
# = - 186.524#
which is our approximation for the integral