Estimate the area under the graph of f(x) = 2 - sin(x)?

Estimate the area under the graph of f(x) = 2 - sin(x)? from x=0 to x=pi using n=6 rectangles and left endpoints/right endpoints/midpoints?

1 Answer
Mar 25, 2018

Exact 5.2831
Approx Upper sum=4.329

Explanation:

Let's estimate using 6 rectangles and a upper sum. Since we are estimating area from 0 to #pi#. Then.
#deltax=(pi-0)/6#
#=pi/6#
Given function is #2-sin (x)#.
So our endpoints be
#(pi/6 , pi/3, pi/2, 2pi/3, 5pi/6 , pi)#
Evaluate each points to get the height of each rectangle.
Evaluating each we get
#(1.5, 1.134, 1, 1.134 , 1.5, 2)#
Adding all this values,
#=8.268#
Multiply by #deltax#
#=8.268×deltax#
#~~4.329#
Where as actual value can be evaluated using integration.
#int_0^pi f (x)dx#
#=int_0^pi (2-sinx)dx#
#=int_0^pi 2dx +int_0^pi -sinxdx#
#=2pi+(cos(pi)#
#=5.283185#
Thank you.