Find the volume of the solid that lies under the hyperbolic paraboloid z = 3y2 − x2 + 3 and above the rectangle  R = [−1, 1] × [1, 3].

The parabola y=(1/2)x² divides the disk x² + y² ≤ 8 into two parts (see figure). Find the area of each part. To solve this problem, perform the following i. Find the points of intersection.  ii. Let A₁ be the region bounded by √(8 - x²) above by the parabola below. (a Trig substitution will be needed) iii. Let A₂ be the region bounded by the parabola above and by the disk below. (area of the disk- area of A₁)

The line x=C where C70 intersects the cubic y= 2x3 + 3x²-9 a七 the point P and parabola y = 4x+ 4x +5 2. at %3D point Q.

Find the distance between the plane (x − 2) +7(y+3)+2z = 0 and the center of the sphere x²+ y²+z²+6x+8y-4x+4=0. Simplify your answer.