Computing the area between 2 curves $x+y = 5$ and  $x+7 = y^2$.
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It is possible to calculate the area between two curves as follows:
$\int_a^b f(x) dx$ + $\int_b^c g(x) dx$. Where a = ___
, b____, c_____ and $f(x) = $________ and $g(x)=$________
Alternatively, this area can be calculated as a single integral:
$\int_z^w t(y) dy$.
Where z = ___
, w____ and $t(y) = $________
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My answers was:
a = -4, b = 0, c = 3. f(x) = 5-y and g(x) = $y^2 - 7
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And z = -4,w = 3 and t(y) = $-y^2 -y + 12$. The value final is: $\cfrac{343}{6}$
I'm correct?
 A: Here is a screengrab of the the two curves and the region between them.  Pardon the not straight lines.

One can describe and calculate the area of the region using vertical bars.  That is what the first half of the problem is trying to get you to describe this problem as using integrals where you integrate over values of $x$.
The area can be described as the area of the blue region which follows one pattern (upper edge of the rectangles lying on the upper part of the parabola) as well as the area of the red region which follows a different pattern (upper edge of the rectangles lying on the line).
So, we try to describe the area as the sum of two integrals each of which with respect to $x$.  To do this, we first ask... what $x$ values correspond to the blue region?  What $x$ values correspond to the red region?

 $-7\leq x\leq 2$ for the blue region.  $2\leq x\leq 9$ for the red region.

Next, given a particular $x$ in the blue region, what is the height of the corresponding rectangle?

 It ranges from the top part of the parabola to the bottom part, i.e. ranges from $+\sqrt{x+7}$ to $-\sqrt{x+7}$.  The height is then the difference of these: $2\sqrt{x+7}$

Given a particular $x$ in the red region, what is the height of the corresponding rectangle?

 It ranges from the top being the line to the bottom being the parabola, i.e. ranges from $5-x$ to $-\sqrt{x+7}$.  The height is the difference of these: $5-x+\sqrt{x+7}$

Finally, we can describe the area using these setting it up as integrals:

 $\int\limits_{-7}^2 (2\sqrt{x+7})\text{d}x + \int\limits_2^9(5-x+\sqrt{x+7})\text{d}x$

The second half of the problem has you dealing with describing the area as an integral with respect to $y$ by effectively adding up the areas of multiple horizontal rectangles.  This is perhaps easier as you don't have to mess with messy square roots and you don't need to break the region up into multiple pieces to handle separately since for every choice of $y$ the width of such a rectangle follows the same pattern.
