Definite Integral and Area

The solution file is attached.

3. Read the statement of this problem very carefully.
The graph of g(x) is given below on the interval [-3,5].

Regions A, B and C are labeled in the graph. The area of region A is 32/3, the area of region B is 32/3, and the area of region C is 32/3. Compute

a )
b)
c )
d )
e )
f ) None of the above.

The given integral gives the total area under the curve. Since area is always positive, we add up the areas to get the total area as 32/3 + 32/3 + 32/3 = 32
But the value of the given integral is -32/3 + 32/3 - 32/3 = -32/3
Option (a)

4. Show all of your work on the answer sheet provided. Make sure you read this problem carefully!
Find the function F which is an antiderivative of

and satisfies F (0) = 1.

F(x) =  f(x) dx = dx/(x + 2)^2 -  2x dx +  3 dx = -1/(x + 2) - x^2 + 3x + C
F(0) = -1/(0 + 2) - 0 + 0 = C  C = -1/2
 F(x) = -1/(x + 2) - x^2 + 3x - ½

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