# Question #ad67f

##### 1 Answer

#### Answer:

The region has an area of

#### Explanation:

I'm going to start at the start of this problem and work my way to the end methodically so that you know exactly what is going on.

We're going to need to find the equation of the tangent line, so we differentiate the parabola's equation (call it

The slope of the tangent is therefore

The equation of the tangent is therefore:

We now compare the two graphs, letting the tangent line be

We now set up our integral:

#=>int_0^5 4x^2 - (40x - 100) dx#

#=>int_0^5 4x^2 - 40x + 100 dx#

#=>int_0^5 4(x^2 - 10x + 25)dx#

#=>4int_0^5 x^2 - 10x + 25dx#

Now integrate using

#=>4[1/3x^3 - 5x^2 + 25x]_0^5#

This can be evaluated using the second fundamental theorem of calculus, which states that

#=>4[125/3 - 125 + 125]#

#=>500/3#

Hopefully this helps!