Estimate the area under the graph of f(x) = 2 + 2x2 from x = -1 to x = 2 using three rectangles and right end-points

Question

john_es · Answer

Well, we choose the height the rectangles to be,
$$f(x) \Rightarrow x \text{  will be the middle point between the x partition }$$
So, our x coordinates will be,
$$x_1=(-1+0)/2=-0.5$$$$x_2=(1+0)/2=0.5$$$$x_3=(2+1)/2=1.5$$
The base of each rectangle will have 1 unit of longitude, because,
$$\text{(End point-First point)/(Number of rectangles)}(2-(-1))/3=1$$
So the area will be,
$$A\approx1\cdot(f(x_1)+f(x_2)+f(x_3))=11.5$$

anonymous · Answer

why do you add 0 to -1 and 1 but add a 1 to 2 in order to find the coordinates ?

john_es · Answer

The idea is to find the middle point of the intervals, where the rectangle is possibly best placed.

john_es · Answer

You can try other heights: left heights (-1,0,1) or right heights (0,1,2), but these sums are a worse aproximation.

john_es · Answer

You can check a graphic example here, http://demonstrations.wolfram.com/ComparingBasicNumericalIntegrationMethods/

anonymous · Answer

Ahh! I see... ok thanks for your help