Question

use the limit process to find the area of the region bounded by the graph of the function f(x)=x+2 and the y axis over the interval of 0 less than or equal to x less than or equal to 2.(divide the interval into n even sub-intervals, and find the sum of the area of rectangles, and take the limit n approaching infinity) check your answer with the geometric formula

Answer 1

Number of sub intervals : n

Let the f(x) be a function and is continuous on the interval a ≤ x ≤ b.

Divide this interval into n equal width subintervals, each of which has a width of   ∆x = (b - a)/n

And so on ,we then have

The Riemann sum is then

 

Using Right Riemann sum, we get :

Area of the rectangle

Area = 2 square units.

 

Answer 2

Number of sub intervals : n

Let the f(x) be a function and is continuous on the interval a ≤ x ≤ b.

Divide this interval into n equal width subintervals, each of which has a width of   ∆x = (b - a)/n

And so on ,we then have

The Riemann sum is then

Using Right Riemann sum, we get :

Area of the rectangle

Area = 6 square units.