How to Find the Area Bounded by Two Complex Curves?

[ProBasket]

Find the area bounded by the two curves:
$$x=100000(5*sqrt(y)-1)$$
$$x=100000(\frac{(5*sqrt(y)-1)}{(4*sqrt(y))})$$

i'm having a lot of trouble trying to find the lower and upper limit of the two functions. I tried setting the two functions together and solving for y, but i get 0. then trying to plug in 0 for y which gives me -100000 for the first function, but you can't plug in 0 for y for the second function.

 

[Davorak]

Well first you should know that you have to solve for y:
$$5 \sqrt{y}-1=\frac{5\sqrt{y}-1}{4\sqrt{y}}$$
so
$$20y-4\sqrt{y}=5\sqrt{y}-1$$
$$20y+1=9\sqrt{y}$$
square both sides and:
$$400y^2+40 y+1=81 y$$
$$400y^2-41 y+1= 0$$
Then use the quadractic equation to find the overlaping area. After that intgrate(one function minus the other) between the two values of y.

Also you could put these equations in terms of y(x) rather then x(y) and follow the same procedure and obtain the same answer.

 

[thepatientmental]

Finding the area bounded by two curves can be a tricky task, especially when the equations are complex. In this case, it seems that you are having difficulty finding the limits of integration for the two curves.

One approach you can take is to graph the two curves and visually determine the intersection points. These points will serve as the limits of integration for your problem.

Another approach is to solve the equations for y and then find the intersection points by setting the two equations equal to each other. However, in this case, it seems that you have already tried this method but encountered some issues.

It is possible that the equations you are working with do not have a common intersection point, which can make it difficult to find the limits of integration. In such cases, you may need to use numerical methods or approximations to find the area bounded by the two curves.

Overall, finding the area bounded by two curves can be a challenging task, but with the right approach and tools, it can be solved. Keep trying and don't hesitate to seek help from a tutor or classmate if needed. Good luck!