Area under a graph (without evaluating integral)

[PaulKraemer]

Hi,

I am having trouble with the following problem that asks you to find the area under the graph of f = sqrt (9 - x^2) from 0 to 3. This problem comes in a chapter before you learn to evaluate integrals. I am following the method used to solve the examples in this chapter, which is basically as follows:

1. Divide the interval [0,3] into n equal subintervals. This makes the length of typical subinterval delta-x = 3/n

x₀ = 0, x₁= delta-x, x₂= 2 * delta-x, x_i = i * delta-x, ... , x_n = n * delta-x = 3

Because delta-x = 3/n, I get...
x_i = i * delta-x = 3i/n

2. Since f is decreasing on [0,3], the number u_i in [x_i-1, x_i] at which f takes on its minimum value is always the right-hand endpoint, that is:

u_i = x_i = 3i/n

3. f(u_i) = f(3i/n) = sqrt(9 - (3i/n)^2)

4. f(u_i) * delta-x = sqrt(9 - (3i/n)^2) * 3/n.

...this is where I get stuck. In all of the examples in this chapter, they solve for the area by calculating the limit as n approaches infinity of the sum from 1 to n of f(u_i) * delta-x.

In all the examples, they simplify this sum by reducing it to sums of i, i^2, or i^3, where they give you the formulas for these sums in terms of n.

None of the examples had the i inside a sqrt. I can't figure out how to reduce the formula for the sum 1 to n of f(u_i) * delta-x to anything I know how to deal with.

Any help will be greatly appreciated. Thanks in advance.

Paul

 

[lookagain]

Headline:

\(\displaystyle \text{Area under a graph (without evaluating integral)}\)

I am having trouble with the following problem that asks you to find
the area under the graph of f = sqrt (9 - x^2) from 0 to 3. This
problem comes in a chapter before you learn to evaluate integrals.


** I am following the method used to solve the examples in this chapter,
which is basically as follows:

PaulKraemer,

seriously, until you mentioned the sentence (**), and the
information that follows it, I would have told you to use a
geometrical method, by looking at


\(\displaystyle \text{the area between the semicircle, }\)
\(\displaystyle {f(x) \ = \ \sqrt{9 - x^2},\)
\(\displaystyle \text{and the x-axis, from x = 0 to x = 3.}\)


This is one of the methods that students I know have worked in
beginning calculus courses.