Question

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?

Answer 1

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.