# How do you find the area between the given curve #y= x^2# and the x-axis given in the interval [0,1]?

##### 1 Answer

The area is

#### Explanation:

I am assuming that you do not yet have the Fundamental Theorem of Calculus available to evaluate this, but that you need to evaluate it from a definition.

.

Where, for each positive integer

And for

I prefer to do this type of problem one small step at a time.

For each

And

# = sum_(i=1)^n i^2/n^3#

# = 1/n^3 sum_(i=1)^n i^2 #

# = 1/n^3[(n(n+1)(2n+1))/6]#

So,

The last thing to do is evaluate the limit as

I hope it is clear that this amounts to evaluating

There are several ways to think about this:

**Limit of a Rational Expression**

The numerator can be expanded to a plynomial with leading term

**OR**

The limit at infinity is

**OR**

# = (1)(1+1/n)(2+1/n)#

So the limit is, again

However we get it, we get

.

# = lim_(nrarroo) sum_(i=1)^n((i^2)/n^2) 1/n#

# = lim_(nrarroo) [1/6[(n(n+1)(2n+1))/n^3]]#

# = 1/6 (2)#

# = 1/3#