Not sure where to even start
Find the area of the region in the first quadrant bounded by the curves y^2=7x, y^2=8x, x^2=4y, x^2=5y
There is four points of intersection between these curves. First thing to do is find them. This will give you the terminals for your integration. Do you know how to do this?
After this apply them to
$\displaystyle A= \int \sqrt{8x}-\frac{x^2}{5}~dx-\int \sqrt{8x}-\frac{x^2}{4}~dx-\int \sqrt{7x}-\frac{x^2}{5}~dx+\int \sqrt{7x}-\frac{x^2}{4}~dx$
Other guys beat me to it. The best thing to do in these programs is to draw out the total region, identify region as defined by our bounds, and then compute the double integral. In any case, the following is how I would solve it
You need to draw these out. We're going to end up with 4 equations in terms of y:
$\displaystyle y = \sqrt{7} \sqrt{x} $
$\displaystyle y = \sqrt{8} \sqrt{x} $
$\displaystyle y = \frac{x^2}{4} $
$\displaystyle y = \frac{x^2}{5} $
This will be a parallelogram in the first quadrent. Notice how
$\displaystyle y = \sqrt{8} \sqrt{x} $ and $\displaystyle y = \frac{x^2}{5} $ intersect at the top
$\displaystyle y = \sqrt{7} \sqrt{x} $ and $\displaystyle y = \frac{x^2}{4} $ intersect at the bottem
So let us bound our function in terms of y and in terms of width (actual values here).
The width (x) can be found by equating the functions that intersect at the top and at the bottem and finding the difference.
So,
$\displaystyle \sqrt{8x} = \frac{x^2}{5} $
$\displaystyle 5 \sqrt{8} = x^{ \frac{3}{2} } \to x_2 = 5.84 $
Then,
$\displaystyle \sqrt{7} \sqrt{x} = \frac{x^2}{4} $
$\displaystyle 4 \sqrt{7} = x^{ \frac{3}{2} } \to x_1 = 4.82 $
We can now construct our double integral.
$\displaystyle A = \int_{4.82}^{5.84} dx \int_{ \sqrt{7x} }^{ \sqrt{8x} } dy $
But wait! In this calculation we have included the sides to the left and to the right of our region. So we must find an expression for these areas so that we can subtract them.
On the left, we have the value $\displaystyle x = 4.82 $ and are extending to the intersection of $\displaystyle \sqrt{8x} = \frac{x^2}{4} $ which leads to $\displaystyle x = 5.03 $
Thus the area THAT WE DONT WANT on the left side is
$\displaystyle A_L = \int_{4.82}^{5.03} dx \int_{ \frac{x^2}{4} }^{ \sqrt{8x} } dy $
Similarly, on the right side we extend from the intersection of $\displaystyle \frac{x^2}{5} = \sqrt{7x} $ which leads to $\displaystyle x = 5.59 $, to $\displaystyle x = 5.84 $
Thus, total area is
$\displaystyle A = \int_{4.82}^{5.84} dx \int_{ \sqrt{7x} }^{ \sqrt{8x} } dy - \int_{4.82}^{5.03} dx \int_{ \frac{x^2}{4} }^{ \sqrt{8x} } dy - \int_{5.59}^{5.84} dx \int_{ \sqrt{7x} }^{ \frac{x^2}{5} } dy $