Find area under y= sinx between [ 0,pi]?(rieman sum)

Question

Find area under y= sinx between [ 0,pi]? . How would I do this using a Rieman sum

Answer 1

I have a question! What kind of sum are you supposed to be doing? You will get a different result (in general) depending on which sum you use (right-hand, center, left-hand, trapezoidal, other).

 

Michael gave you the solution for a right-hand sum. If you were using left-hand sums (for example), then you would have:

   Area under curve = (π/4)[f(0) + f(π/4) + f(π/2) + f(3π/4)]

Answer 2

Remember drawing small rectangles under the curve at small but equal intervals of x?  Then finding the areas of those rectangles and adding up their areas?  That is the Riemann sum.

 

∑f(x)Δx = sum of the sub-areas under the curve.

 

 

First, lets find a length for the base of the rectangles.  We can do that by find taking the sum of the interval and dividing the result by the number of rectangles we want to use.

 

(0 + π) / 4 = π/4

 

This means the length of each rectangle base is π/4 using 4 rectangles.  The height of the rectangles is f(x).

 

Now using Riemann's sum,

 

Area under curve = (π/4)[f(π/4) + f(π/2) + f(3π/4) + f(π)]