Question

Area of the Region Between Curves-Please help.

Answer 1

Answer: 32/3

Step-by-step explanation:

To find the area of the region between the curves, you first want to find the interval of both curves. You can do that by setting both curves equal to each other.

$-\frac{1}{2}x^2+2=\frac{1}{2} x^2-2$                   [subtract both sides by 2]

$-\frac{1}{2}x^2= \frac{1}{2} x^2-4$                         [subtract both sides by $\frac{1}{2} x^2$]

$-x^2=-4\\x=2,-2$        

Now that we know the interval, we set this into an integral and subtract the top curve by the bottom curve. *Note: I can't type in -2 into the interval, therefore only a "-" will be displayed.

$\int\limits^2_- {|-\frac{1}{2}x^2+2-(\frac{1}{2}x^2-2) | } \, dx$                     [distribute -1]

$\int\limits^2_-{|-\frac{1}{2}x^2+2-\frac{1}{2}x^2+2 |} \, dx$                       [combine like terms]

$\int\limits^2_- {|-x^2+4|} \, dx$                                          [solve integral]

$\frac{32}{3}$

The area between the curves is 32/3. This was also the reason why we had the absolute values because area can never be negative.