2. Compute the volume integral of V fin the (2) unit ball for f (r 1.5a)/r 1.5al3
2. Geometric interpretation of integrals. Consider the integral where R is the region bounded by the a-axis, p-axis and r +y- 2 (a) Let =-z-v + 2, what object does this equation (NOT the integral) represent? (b) Interpret the integral as the volume of a shape. Sketch the shape. (e) Compute the integral by computing the volume of the shape. Page 3
2. Geometric interpretation of integrals. Consider the integral where R is the region bounded by the a-axis, p-axis...
dV, where is the unit ball in R3, that is, Use spherical coordinates to compute the integral We E = {(x, y, z)| 22 + y2 + 2 <1}.
1. Use the definition of integral to show that if f : B → R and f are integrable, then Inf e f 2. Find the volume of the region K between ~ = x2 + 9y2 and z = 18-x2-9y2 (Use Fubini's Theorem) 3. Evaluate Jryz where S is the upper half of the unit sphere. (Use Change of Variable Theorem)
Let F be the vector field on R3 given by F(x,y,z)=(2xz,-x,y^2)
evalute the volume integral below. cheers
19. Let F be the vector field on R given by F(r,y,z) = (2xz, -x, y2) Evaluate 2xzdV, FdV xdV where V is the region bounded by the surfaces 0, y = 6, z = x2 and z = 4. 0, y
Section 2.9 I. Set up a double integral to compute the volume of the solid under the curve : = r2-8 bounded by 0 1 and 0 V 2. Then find the volume of the solid.
(b) Compute (f. V)f, where f is the unit vector defined in Eq. 21,
(2) Suppose V F = (ex: -2yre?",0). Compute SS (V F). ds, where is the upper unit hemisphere r? + y + z2 = 1, 2 > 0. (Hint: Can you use result of 1 and a more convenient surface over which to integrate?)
(b) Let F(r) be a force vector so that the line integral F(r)r is the work W done by F(r) along the curve C with parametric representation r(r) which is the displacement vector. Let m be the mass of the object. Prove that the work done on the object within te[to4 satisfies the equation W- cF(r).dr-îm v(r 2Г which states the work energy theorem: work done on an object equals to the change of kinetic energy (Hint I: Newton's second...
consider a spherical ball of of charge radius R with a volume charge density of p(r)=a^3 for r≤R what are coefficient unit, calculate the electrical field r≥R and show that the expression agrees when r=R
1. Consider the unit circle: (x,y) : x2 y2 = 1. T. Let f R2 ->R be defined by f(x,y) = x2-y, and let F : R2 -> R be defined by F(x, y) Compute the integral of f and F around the unit circle. For the integral of F, proceed in the standard (anticlockwise) direction