x
1. Evaluate the iterated integrals: x2+2x+y a. JR 3x+3y dA, R: 15x32,0 sys 1 (Hint: Simplify...
Evaluate the triple integrals JR V and JSSR zdv, where R is the region bounded above by the sphere x2 +y2+22 : 4, below by the cone 3za_ x2 + y2, and such that y 2 0 Evaluate the triple integrals JR V and JSSR zdv, where R is the region bounded above by the sphere x2 +y2+22 : 4, below by the cone 3za_ x2 + y2, and such that y 2 0
QUESTION 4 Use the appropriate transformation to evaluate SX (2x + y)(x - y)dA where R is the region bounded by the line y = 4 - 2x, y = 7 - 2x, y = x - 2 and y = x +1. (8 marks)
1) Given the following iterated integral. ex/Y DA R = y = 4x, y = -xy = 8 a) (0.75 point) Sketch the bounded region R. Label your graph. b) (1.25 point) Evaluate the definite integral with the given function over the bounded region R.
3. (A) (Change of Variables) Evaluate the following integrals by making appropriate change of variables. (a) // sin(x2 + y2) dA, where R is the region in the first quadrant bounded by the circle x2 + y2 = 5. YdA, where R is the parallelogram enclosed by the four lines 3. -Y x - 2y = 0, 2 - 2y = 4, 3.x - y = 1, and 3.c - y = 8. zevky / dA, where R is the...
Evaluate the integral 1 1 33x2 +3y? da JR where the region R is given by the figure with a = 3 and b = 5. (Assume the curved boundary of the figure is circular with center at the origin.) SUR į V3x2 + 3y2 dA =
3. Use the transformation u = xy, v = y to evaluate the integral ∫∫R xy dA, where R is the ay region in the first quadrant bounded by the lines y = x and y = 3x, and the hyperbolas xy = 1, xy = 3
Exercise 6. (17pts) In this exercise use double integrals. a. Evaluate the integralj"fo/ b. Find the volume of the solid whose base is the region R in the ry-plane bounded by the curve y --x? +2x and the line y - x-2, while the top of the solid is bounded by the surface z xy e" Exercise 6. (17pts) In this exercise use double integrals. a. Evaluate the integralj"fo/ b. Find the volume of the solid whose base is the...
Use the transformation u = 3x + y, v=x + 3y to evaluate the given integral for the region R bounded by the lines y = - 3x + 1, y= - 3x + 3, y= - = X, and y=- -x + 2. ne lines y = – 3x+1, y = – 3x+3, y=-3x, and y=-**+2. 3 Siſ(3?+ 16 +3%) dx ay SJ (3x? + 10x9 +35) dx dy=0 (Simplify your answer.)
6. (4 pts) Consider the double integral∫R(x2+y)dA=∫10∫y−y(x2+y)dxdy+∫√21∫√2−y2−√2−y2(x2+y)dxdy.(a) Sketch the region of integration R in Figure 3.(b) By completing the limits and integrand, set up (without evaluating) the integral in polar coordinates.∫R(x2+y)dA=∫∫drdθ.7. (5 pts) By completing the limits and integrand, set up (without evaluating) an iterated inte-gral which represents the volume of the ice cream cone bounded by the cone z=√x2+y2andthe hemisphere z=√8−x2−y2using(a) Cartesian coordinates.volume =∫∫∫dz dxdy.(b) polar coordinates.volume =∫∫drdθ. -1 -2 FIGURE 3. Figure for Problem 6. 6. (4 pts)...
7. Evaluate the following integral by converting to polar coordinates: S], 127 (2x – y)dA, where R is the region in the first quadrant enclosed by the circle x2 + y2 = 4 and the lines x = 0 and y = x. 8. Find the surface area of the portion of the plane 3x + 2y +z = 6 that lies in the first octant. 9. Use Lagrange multipliers to maximize and minimize f(x, y) = 3x + y...