I need some help on this Calculus question about polar coordinates.

Consider a pasture bordering on a straight river. At 2m from the river, a goat is tied to a pole with a 4m long rope. What is the area of grass that the goat can graze?

There are a couple ways to set this up, but I think it should be a circle of radius 4 at the origin. Let the river run north-south along the line x=2 or r*cos(theta)=2
My graphics application got approx. 40.4384m^2, use that for a check on your integral.

I don't know how much my first post to your question helped, so I'll try again.
Here are some suggested steps to try.
(1) Draw a circle of radius 4 centered at (0,0).
(2) Draw the line x=2
(3) Shade the area to the right of the line and inside the circle.
Use this diagram as a guide. Now we're interested in the upper half of the circle and take advantage of the symmetry.
(4) If you draw a segment from the point of intersection of the line to the origin, it should make an angle of pi/3 with the x-axis.
There is now a triangle formed by the line x=2, the x-axis and the segment from (4).
You should be able to determine the area of the sector and the triangle separately in polar coordinates.
The limits for the triangle go from 0 to pi/3 and those for the circle from pi/3 to pi.
Remember to multiply the area by 2 to get the total area.
I hope this is more helpful, if not please repost.

For step 3), why do u shade the area to the right....from the diagram that i drew, i thought it should be to shade the area to the left?

I was think of that as the river, but you're correct, the area to the left of that line is the area of integration and that should be indicated as such.

What coordinates are 33 30S (LATITUDE) and 70 40W (LONGITUDE)?

To determine the coordinates 33 30S (latitude) and 70 40W (longitude), we can use the following conventions for latitude and longitude:

- Latitude is measured in degrees north (N) or south (S) of the equator. The equator is at 0 degrees latitude.
- Longitude is measured in degrees east (E) or west (W) of the Prime Meridian, which passes through Greenwich, London. The Prime Meridian is at 0 degrees longitude.

Given that the latitude is 33 30S, it means it is 33.5 degrees south of the equator.
Similarly, the longitude 70 40W means it is 70.67 degrees west of the Prime Meridian.

So, the coordinates 33 30S and 70 40W represent a location that is 33.5 degrees south of the equator and 70.67 degrees west of the Prime Meridian.