Find the area of the region

Question

Decide whether to integrate with respect to x or y. Then find the area of the region.

2y=5√x. y=5, and 2y+1x=6

 

Answer 1

http://www.wolframalpha.com/input/?i=graph+y%3D5%2C+y%3D5%2F2sqrt%28x%29%2C+and+y%3D3-x%2F2

 

As you can see from the Wolfram Alpha plot, our region is in the first and second quadrants. If we were to integrate with respect to x, i. e.

 

∫ ---------- dx

 

then our boundaries would have to be in terms of x. Moving across the region from left to right, I can see that while my upper bound stays the same throughout, my lower bound switches at x=1, so I would have to split the region into two pieces (to the left and right of x=1). That means two separate integrals.

 

However, if I integrate with respect to y, i. e.

 

∫ ---------- dy

 

then I have constant functions for my lower and upper bounds. That means one integral to find the whole area. Nice! We should definitely integrate with respect to y to make our lives easier.

 

In this case, the upper bound with be the higher values of y, or the parabola. The lower bound would then be the lower values of y, or sloped line.

 

You should have enough information to set up and work out the integral by yourself now. I'll tell you that the final answer is about 9.6, but I'm sure you'll have to show your work to get there!