Calculus Question?

Question

Write a limit to find the area of the region bounded by the curve y=x^2, the positive x axis, and the line x=3. Then find the area. 

I have no idea how to approach this problem, please help!

Answer 1

Since ,  use limits, then we should probably use a Riemann sum and calculate as n → ∞.

Interval : x = 0 to 3(i.e, b = 3 and a = 0)

Number of subintervals : n

Let f (t) be a function that is continuous on the interval a ≤ t ≤ b. Divide this interval into n equal width subintervals, each of which has a width of .

Subintervals width : .

For each subinterval, we have rectangle with width = and height = f(x), where f(x) = x²

Using Right Riemann sum, we get :

Area of the rectangle .

 

 

 

 

 

To find exact area, take limit as n → ∞.

 

 

 

 .

Therefore, area of the region bounded by the curve y = x², the positive x - axis and the line x = 3 is 9 square units.