Set both parabolas equal to each other and solve for x.

Substitute both values of  into either parabola and determine .

The coordinates of intersection are:

 and 

To solve, set both equations equal to each other:

To solve as a quadratic, combine like terms by adding/subtracting all three terms from the right side to the left side:

This simplifies to

Solving by factoring or the quadratic formula gives the solutions and .

Plugging each into either original equation gives us:

Our coordinate pairs are and .

Evaluating this integral requires use of the "Product to Sum Formulas of Trigonometry":

For:

So for our given integral, we can rewrite like so:

This can be rewritten as two separate integrals and solved using a simple substitution.

Solving each integral individually, we have:

Substituting this into the integral results in:

The other integral is solved the same way:

Substituting this into the integral results in:

Now combining these two statements together results in one of the answer choices:

This integral can be easily evaluated by following the rules outlined for integrating powers of sine and cosine.

But first a substitution needs to be made:

Now that we've made this substitution, we will use the rules outlined for integrating powers of sine and cosine:

In General:

1. If "m" is odd, then we make the substitution , and we use the identity .

2. If "n" is odd, then we make the substitution , and we use the identity .

For our given problem statement we will use the first rule, and alter the integral like so:

Now we need to substitute back into v:

Now we need to substitute back into u, and rearrange to make it look like one of the answer choices:

Step 1: Draw a  triangle..

The short sides have a length of  and the hypotenuse has a length of .

Step 2: Find Sin (Angle A):




Step 3: Rationalize the root at the bottom:

The inverse trig functions should be memorized.

The other common inverse trig functions are

Step 1: See if we can plug in  into the equation..

We can't because the denominator becomes ..

Step 2: Factor the denominator:

 (By the Difference of Perfect Squares Formula)

Step 3: Re-write the function:



Step 4: Divide by  on both the numerator and denominator because it's common:

We are left with: 

Step 5: Plug in :



The limit of this function as x approaches 2 is .

Let's examine the limit

first.

and 

,

so by L'Hospital's Rule, 

Since ,

Now, for each , ; therefore, 

By the Squeeze Theorem, 

and 

Therefore, by L'Hospital's Rule, we can find  by taking the derivatives of the expressions in both the numerator and the denominator:

Similarly, 

So 

But  for any , so 

and 

Therefore, by L'Hospital's Rule, we can find  by taking the derivatives of the expressions in both the numerator and the denominator:

Similarly, 

so