The region bounded below by the line Y equal -3 and above by the parabola Y equal negative X ^2 + 1 is to be partitioned into two subsections of equal area by cutting across with the horizontal line y = C Find C by integrating with respect to Y. This puts C in the limits of integration. So the first thing we need to do is we need to find the area that's actually being enclosed so that then we can figure out how to split it in half so we have two equal areas. So we're going to start by setting the two equations equal to figure out where they're intersecting -3 is going to equal negative X ^2 plus One X ^2 - 4, so X + 2 X -2 negative. 2:00 to 2:00 We realized we have symmetry here, so I'm going to do 2 from zero to two in the X direction. So we're going to get the upper equation, IE the negative X ^2 + 1 minus the lower equation, so minus a -3. So two from zero to two of negative X ^2 + 4 DX. Evaluating that integral, we get two negative one third X ^3 + 4 X. Putting in our upper and lower bounds and computing, we see we get 32 thirds units squared. So 2 equal areas would have each of the areas being 16 thirds in each of the area. So now we're going to actually find the integral and we want to do it in terms of Y. So we're going to do DY. So we're going to end up having horizontal lines. So we're going to go to -3 to C, the square root the furthest to the right, which is if we solve for X, we get sqrt 1 -, y -, r zero. I'm going to multiply by two due to the symmetry, and that's got to equal my 16 thirds. So we're going to take that sqrt 1, or we're going to take the square root, add 1 to it, and get three halves. Divide by the new one, which is 2/3. But then by the chain rule, we have to remember the derivative of the inside was -1. So we're going to get negative 2/3 the quantity 1 -, y to the three halves. I'm going to take the two that was out in front and divide it over to the other side. So instead of 16 thirds, we're going to have 8 thirds. Putting in our upper bound of C and our lower bound of -3. We get 1 -, C to the three halves -4 to the three halves equaling. I'm going to multiply the negative 2/3 to the other side now, so equal -4 4 to the two third, or four to the three halves. Sqrt 4 is 22, cubed is 8, so I'm going to take that -8 and add it to the other side. Then I'm going to take each side to the 2/3 power. So we get 1 -, C equaling 4 to the 2/3. Solving for C, we get C equaling 1 - 4 to the 2/3. We could have also done the integration if we had wanted to, from the lower value of Y being C to the upper value of Y being one. If we did two of those, we'd still go from the furthest to the right function minus the further to the left function, equaling the 16 thirds. If we compute that out, we'd still get 1 -, 4 to the 2/3.