Bryce P. answered • 12/19/19

CRLA Certified Mechanical Engineering Tutor w/ 1,500 hrs of experience

Hi Isaac,

__Sub-part i and ii__

For graphing I suggest you use Desmos Online Graphing Calculator. By graphing these two equations you can visualize the bounded area and note that *y*_{2}* = x + 5* is the top bound of the area and *y*_{1}* = 2x*^{2}* + 2 *is the bottom bound.

__Sub-part iii__

To solve for the intersection points of two lines simply set the two equations equal to one another and solve.

*2x*^{2}* + 2 = x + 5*

*2x*^{2}* - x - 3 = 0*

*(2x - 3)(x+1) = 0*

The two lines would thus intersect at *x = 3/2* and *x = -1*. Use the graph to verify that this is correct.

__Sub-part iv__

To solve for the area between two curves use the formula

*∫*_{a}^{b}* f(x) - g(x) dx *

where f(x) is the top bound of the area, *g(x)* is the bottom bound of the area, and *[a,b]* is the interval over which we are solving for the area.

In this case we know f*(x) = x + 5, g(x) = 2x*^{2}* + 2*, and *[a,b] = [-1, 3/2] *so

*∫*_{-1}^{1.5}* x + 5 - (2x*^{2}* + 2) dx*

*∫*_{-1}^{1.5}* x + 5 - 2x*^{2}* - 2 dx = 5.20833*

I hope this helps! If you have any questions let me know.

Best of luck,

Bryce