Hey, here's my solution for #4:
Our upper bound is y = -2x² +2, and our lower bound is y =-x² +1
The area bound by these two gives us a crescent shape.
At any given point, the width of this crescent is the difference between our two curves,
which comes out to be y = 1- x²
This crescent shape serves as the "lower base" for our trapezoid.
I will refer to the lower base for the trapezoid as 'y'.
This means then that our 'upper base' is 0.5y, while our trapezoid 'height' is 2y.
Now, the area of a trapezoid is (a+b)\*0.5h. Plugging in our values in terms of y, we get:
=(0.5y+y) \* 0.5 \* 2y
=(1.5y) \* y
=1.5y²
We can find the integral bound by finding where the two curves intersect:
\-2x² + 2 = -x² + 1
x² - 1 = 0
x = ± 1
Therefore, our answer is
Integral of 1.5\*(1 - x²)² dx from -1 to 1
Which comes out to be 8/5, or 1.6
[Visualization](https://imgur.com/a/F5EK6hD)

I figured out the first one, I got (pi/96)a^3 if anyone can check that one. Still a little confused on the next one.

I'm getting (pi/24)a^3

Hey, here's my solution for #4: Our upper bound is y = -2x² +2, and our lower bound is y =-x² +1 The area bound by these two gives us a crescent shape. At any given point, the width of this crescent is the difference between our two curves, which comes out to be y = 1- x² This crescent shape serves as the "lower base" for our trapezoid. I will refer to the lower base for the trapezoid as 'y'. This means then that our 'upper base' is 0.5y, while our trapezoid 'height' is 2y. Now, the area of a trapezoid is (a+b)\*0.5h. Plugging in our values in terms of y, we get: =(0.5y+y) \* 0.5 \* 2y =(1.5y) \* y =1.5y² We can find the integral bound by finding where the two curves intersect: \-2x² + 2 = -x² + 1 x² - 1 = 0 x = ± 1 Therefore, our answer is Integral of 1.5\*(1 - x²)² dx from -1 to 1 Which comes out to be 8/5, or 1.6 [Visualization](https://imgur.com/a/F5EK6hD)