Basic Q on definite integrals and areas under the curve

[JOhnJDC]

Homework Statement

Consider the function f(x)=x² on the interval [0,b]. Let n be a large positive integer equal to the number of rectangles that we will use to approximate the area under the curve f(x)=x². If we divide the interval [0,b] by n equal subintervals by means of n-1 equally spaced points, then we have point x₁=b/n, x₂=2b/n, ..., x_n-1=(n-1)b/n.

From this, I understand that the base of each rectangle will be b/n. What I'm not clear on is how to determine the heights of each rectangle. My book says that if we use the upper sums (rectangles that reach just above the curve), then the heights are f(x₁)=(b/n)², f(x₂)=(2b/n)², ..., f(x_n)=(nb/n)².

Can someone please explain? I can't consult a professor or other students because I'm studying calculus on my own. Thanks.

 

[tiny-tim]

Hi JOhnJDC!

You're approximating, so you can take the maximum height, the minimum height, or an average height.

If you take the minimum height, you use the height at n, if you take the maximum height, you use the height at n+1.

(and in the limit, of course, it makes no difference anyway)

 

[LCKurtz]

Your parabola is increasing on [0,b]. So if you look at the two vertical sides of any rectangle, the side on the right will hit x² at the highest point. So the rectangle based on [(i-1)/n,i/n] will be tallest if you use the right end on the function: (i/n)² for its height, and the shortest if you use the left end:((i-1)/n)².

 

[JOhnJDC]

Thank you both. I see it now.