Approximate the area under the curve

Question

Approximate the area under the curve ³ from  to  using a Right Endpoint approximation with 6 subdivisions.

Answer 1

The width of each sub-interval will be (3-0)/6 or w=1/2.

The list of x_i will start at right hand end point of first sub-interval, i.e. 1/2 then increase by 1/2 for the next right hand endpoint.

x₁=1/2

x₂=1/2 + 1/2 = 1

x₃=1/2 + 2(1/2) = 3/2

x₄=1/2 + 3(1/2) = 4/2 = 2

x₅= 1/2 + 4(1/2) = 5/2

x₆= 1/2 + 5(1/2) = 6/2 = 3

The height of each rectangle will be (x_i)³.

The area of the ith rectangle will be (x_i)³w.

So R₁ has an area = (1/2)³(2)

R₂ has an area (1)³(2)

And so on.

Add up the areas of the 6 rectangles and you will have an approximation to the area.

Try visiting this graph to see the result.:

desmos.com/calculator/fthc1f9d8o