Use Green's theorem to calculate the line integral over the curve C of the function sine parentheses x squared close parentheses dx plus open parentheses 3x minus y close parentheses dy where C is a right triangle with vertices three points negative 1 comma 2, the point 4 comma 2, and the point 4 comma 5 oriented counterclockwise.

Now Green's theorem says that we can turn this line integral into the double integral over a region d with respect to a of the function qx minus py. Now in this case, our p is sine parentheses x squared close parentheses and our q is parentheses 3x minus y close parentheses. So we'll take those partials, first off.

So here qx would be equal to 3 and p sub y would be equal to 0. So this is going to become in a rule over the region d with respect to a of 3 minus 0 or 3. So in order to find this line integral, we need to figure out what the region, the boundaries on the region d is going to be. Well, imagine what those points look like. In fact, I will draw those points on a coordinate axis.

So the point negative 1 2, the point 4 2, and the point 4 5. These, of course, from a right triangle. It's up to us to figure out what the boundaries are. Well, I'm going to go ahead and save the boundaries on x are from negative 1 to 4. So x is the boundaries negative 1 to 4, but that makes y-- that boundary is a function of x. And so if we find the slope and the y-intercept for the line connecting the points negative 1 2 and the point 4 5, we see that these boundaries go from 2 to lower boundary up to 3/5 x plus 13/5. Writing that in terms of x.

Sorry, a double integral over the region d with respect to a of 3. It's going to be equal to the integral from negative 1 to 4 with respect to x of the integral from 2 to 3/5 x plus 13/5, with respect to y, of the function 3. Now because this is a constant this actually is equal to 3 times the area of the triangle. And it turns out that the area of that triangle is-- well, if it a base of 5 and a height of 3. So that's going to be 15 over 2. So that this total line integral value is 45 over 2.