Example// Find the volume enclosed by two surfaces Z1=2+x² +y2 ,Z2=4-x² - y² Solution // Z1 Z2 2+x² + y² = 4x² - y² 2 + x² + y² - 4 + x² + y² = 0 2x²+2y2 2 0 x² + y² = 1 ↔ x² + y² = p²² y=√1-x² Y2=-√√1-x= Volume=ff zdA=ff z dy dx = ff (Z2 - Z1) dy dx = (4 x2 y2-2-x² - y²) dy dx Volume-2 √1-2 (2- 2x² - 2y²)dy dx Or Volume-(2- 2x² – 2y²)dy dx Or Volume-41-2 ( (2-2x²-2y2)dy dx v=4 - 4 √² (2y – 2x² y — ²x²³)| √1-x² dx - = 4 + ſo (2√1 − x² - 2x² (1 − x²) - -2x²(1-x²) 2 (√1 = x²)³) dx 2(√1-x2)3 - 3

find volume of tetrahedron with vertices W= (2,2,2)  X= (5,4,3) Y= (5,4,4) Z= (5,5,5)

Find the enclosed area bounded by the intersection of: y = −x^2 + 4x + 1 and y = x + 1.

Find the area of the curved surface z=x+y^2 on a triangle with three points (0,0), (1,1), and (0,1) as vertices.

Cavalieri's Principle states that if a family of parallel planes gives equal cross-sectional areas for two solids S, and S2, then the volumes of S1 and S2 are equal. (a) Does the following proof of Cavalieri's Principle look reasonable? Volume(S1) = ^ A(z)dz = Volume(S2) since the cross-sectional area A(z) at height z is the same for both solids. O yes no (b) Consider the oblique cylinder shown in the figure, with radius of the base r = 2 and height h = 2. h Use Cavalieri's Principle to find the volume V of this oblique cylinder. V =

Find the area of part of sphere x² + y² + z² = 100 that lies inside x² + y? = 10x.