1 Home Quiz 2, MATH 251, Section 505 Due April, 7th 2016 last name : . . . . . . . . . . . . . . . . . . first name : . . . . . . . . . . . . . . . . . . ”An Aggie does not lie, cheat or steal, or tolerate those who do” signature : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Write up your result, detail your calculations if necessary and BOX your final answer. Your final work have to be neat, so use pencil first if you want. Z 1. [8pts] Evaluate the integral by reversing the order of integration 0 8Z 2 1 y3 p x4 + 1 dx dy. 2. [15pts] Find the volume of the solid E in the first octant bounded by the paraboloid z = 12 (x2 + y 2 ) p and the cone z = 2x2 + 2y 2 and coordinate planes. 3. [10pts] Evaluate the integral x2 + y 2 = 1 and x2 + y 2 = 2. RR D xydA where D is the region of the first quadrant that lies between 2 Z π Z 4. [8pts] Sketch the solid whose the volume is given by 0 0 π 4 Z 1 ρ2 sin φdρdθdφ. 0 5. [8pts] Find the volume of the solid E bounded by the paraboloid z = 2 − x2 − y 2 and the paraboloid z = 1 + x2 + y 2 . 6. [15pts] Find the area enclosed by the ellipse x2 y 2 + = 1 (Hint : Use Green’s Theorem). 2 3 3 7. Let the tetrahedron be bounded by the coordinates planes and the plane 3x + 3y + z = 1 with density ρ(x, y, z) = y . (a) [8pts] Find the mass of this tetrahedron. (b) [8pts] Find the y-coordinate ȳ of the center of mass of this tetrahedron. Z 8. Consider the line integral F.dr where F is the vector field F =< 4x2 − 6y, 4x + y 2 > and C is the C positively oriented circle of radius 3 centered at the origin. Evaluate the integral (Show your work) (a) [10pts] Directly (b) [10pts] By Green’s Theorem 4 LAST NAME : FIRST NAME :