1) Use the Substitution Formula to evaluate the integral.

I'm doing my best looking at my lecture notes and examples from the book to guide me into the procedure of substituting, but its not helping me. Could someone show me some light on how to get this started?

2) Find the area of the regions enclosed by the lines and curves given below.

y=x2 - 4xandy= 6x

so I graphed the two functions, the limits of integration for the left hand most region is a = 0 and b = 4 . to solve for the right hand limit, I set the equations equal to each other, and got b=10.

For $\displaystyle 0 \leq x \leq 4 : f(x) - g(x) = 6x - 0 = 6x$

For $\displaystyle 4 \leq x \leq 10 : f(x) - g(x) = 6x - (x^2 - 4x) = -x^2 +10x$

Add the Area of both regions:

Total Area = $\displaystyle \int^4_0 6x dx + \int^{10}_{4} (-x^2 + 10x) dx$

Integrate

$\displaystyle

[3x^2]^4_0 + [\ \frac{-x^3}{3} + 5x^2 \ ]^{10}_{4} $

Solve:

$\displaystyle 3(4)^2 - 0 + ( \frac{-(4)^3}{3} + 5(4)^2) - ( \frac{-(10)^3}{3} + 5(10)^2)$

$\displaystyle

48 - 0 + \frac{-64}{3} + 80 - ( \frac{-1000}{3} - 500)

$

$\displaystyle

\frac{144}{3} - \frac{4}{3} + \frac{240}{3} + \frac{1000}{3} - \frac{1500}{3}

$

$\displaystyle \frac{-120}{3} = -40$

Where did I screw up? It seems right to me...