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2. Consider the segment with endpoints (0,r) and (h.R), where R>r> 0 and h> 0. Find...
Instructions: Show all your work for FULL credit. Calculators are NOL final answer. Neatness is highly appreciated. 1. A region R, bounded by y 2x, y 6-x, and x-axis, is rotated around the y-axis. Sketch the region R, in the box a) 15 strip/slice you will use to find the volume of the solid of revolution. b) Write the definite integral that gives a X the volume of the solid of revolution. (DO NOT evaluate the integral.) Find the circumference...
2. Consider the line segment & the zz-plane given by cone C by rotating the line about the z-axis. fromz 0 to z4. We can then obtain a 2 (a) 4 pts Find a parametric representation r(u, v) for C, including bounds for u and v. (b) (4 pts Calculate and simplify r x rl (c) 3 pts Use a double integral to find the surface area of C 2. Consider the line segment & the zz-plane given by cone...
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...
cannot figure out how to write the integrals for this problem #2 1. If glx) -2x and fx) - , find the area of the region enclosed by the two graphs. Show a work for full credit. (4 pts) 2. A:12-80% 3 3 2 Let fix)-. Let R be the region in the first quadrant bounded by the gruph of y - f(x) and the vertical line x # l, as shown in the figure above. (a) Write but do...
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)...
3. Let region R be bounded by y = 2x - x? and y = 0 on (0,2). Setup the definite integral(s) that represents the volume of the solid generated by rotating region about the y-axis. Draw a sketch to interpret your results.
4. (Calculator) Let R be the region bounded by the graphs of f(x)= 20+x-x2 and g(x)=x-5x. (a) Find the area of R. (b) A vertical line x k divides R into two regions of equal area. Write, but do not solve, an equation that could be solved to find the value of k (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are isosceles right triangles with the hypotenuse...
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 integrationRin Figure 3.(b) By completing the limits and integrand, set up (without evaluating) the integral in polar coordinates. -1 -2 FIGURE 3. Figure for Problem 6. 6. (4 pts) Consider the double integral V2 /2-y² + = (x2 + y) dx dy + + y) do dy. 2-y2 (a) Sketch the region of integration R in Figure 3. (b) By completing the limits and integrand, set up (without evaluating)...
2) The region R in the first quadrant of the xy-plane is bounded by the curves y=−3x^2+21x+54, x=0 and y=0. A solid S is formed by rotating R about the y-axis: the (exact) volume of S is = 3) The region R in the first quadrant of the xy-plane is bounded by the curves y=−2sin(x), x=π, x=2π and y=0. A solid S is formed by rotating R about the y-axis: the volume of S is = 4) The region bounded...
12 3. (10 points) A region R is bounded by the lines x plot of R is shown below. unded by the lines x = 1, r = 2, y = 0, and y = x2. A (a) (5 points) Set up a definite integral to calculate the volume of the solid formed by revolving R around the x-axis. (b) (5 points) Evaluate your integral.