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Q5 Polar Land Areas 30 Points A large plot of land must be equally divided between...
2/6 2" (10%) (a) Sketch the graphs of the two functions y=z(4-z) and y=z and mark the finite region (R) enclosed between them. (Identify (R) carefully!) (b) Let W be the volume of the solid o...
2/6 2" (10%) (a) Sketch the graphs of the two functions y=z(4-z) and y=z and mark the finite region (R) enclosed between them. (Identify (R) carefully!) (b) Let W be the volume of the solid of revolution obtained by revolving the above region (R) around the y-axis. (i) Use the shell method to write down the integral for W. (No need to evaluate the integral.) (ii) Repeat part(i) by using the disk method.
2/6 2" (10%) (a) Sketch the graphs...
12 3. (10 points) A region R is bounded by the lines x plot of R...
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.
8) The part of the curve y = ex + e- x/2 between the points A...
8) The part of the curve y = ex + e- x/2 between the points A (0,1) and B (1, e2 +1/2e) is given. a- Take and edit the derivative of the given function. b- Write and edit the integral that gives the surface area of the object formed by rotating the given part around the y axis. (Hint = write the integral according to x.) C-Solve the integral.
2. Evaluate the following indefinite integrals: (a) vel V=(x+2) dx ET (b) 3. Evaluate the following...
2. Evaluate the following indefinite integrals: (a) vel V=(x+2) dx ET (b) 3. Evaluate the following definite integrals: (a) cos(x) da (sin(x) +18 (b) COS 4. The graph of y=g(t) is shown below, and consists of semicircles and line segments. y=g() -1 3 6 596 s(t) dt Define the function f(x) by f(x)= Use the graph of y = g(t) and the properties of the definite integral to find: (a) the value of (i) f(3) (ii) f(-1) (iii) 1'(6) (b)...
7. Evaluate the following integral. b) SO3(2-1)05 dz c) f (xl dx (15 marks) 8. A...
7. Evaluate the following integral. b) SO3(2-1)05 dz c) f (xl dx (15 marks) 8. A shaded region S is bounded by the curve y = the x-axis, the vertical lines x = 1 and x = V2. The x(4-2)1/4 region is rotated through 21 radians about x-axis to form a solid of revolution. Compute the exact volume of the solid of revolution formed. (10 marks) 9. Between the two towers of a suspension bridge that are 2 kilometres apart,...
1. (13 points) Use the limit of a Riemann Sum (i.e. sigma notation and the appropriate...
1. (13 points) Use the limit of a Riemann Sum (i.e. sigma notation and the appropriate summation formulas) to evaluate the net-signed area between the graph of f(0) = 23 – 3 and the interval (0, 2). Let 27 be the right endpoint of the k-th subinterval (where all subintervals have equal width). Give your answer as a single integer or frac- tion, whichever is appropriate. Using any technique other than a limit of a Riemann Sum will earn no...
2/6 2" (10%) (a) Sketch the graphs of the two functions y=z(4-z) and y=z and mark the finite region (R) enclosed between them. (Identify (R) carefully!) (b) Let W be the volume of the solid of revolution obtained by revolving the above region (R) around the y-axis. (i) Use the shell method to write down the integral for W. (No need to evaluate the integral.) (ii) Repeat part(i) by using the disk method.
2/6 2" (10%) (a) Sketch the graphs...
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.
8) The part of the curve y = ex + e- x/2 between the points A (0,1) and B (1, e2 +1/2e) is given. a- Take and edit the derivative of the given function. b- Write and edit the integral that gives the surface area of the object formed by rotating the given part around the y axis. (Hint = write the integral according to x.) C-Solve the integral.
2. Evaluate the following indefinite integrals: (a) vel V=(x+2) dx ET (b) 3. Evaluate the following definite integrals: (a) cos(x) da (sin(x) +18 (b) COS 4. The graph of y=g(t) is shown below, and consists of semicircles and line segments. y=g() -1 3 6 596 s(t) dt Define the function f(x) by f(x)= Use the graph of y = g(t) and the properties of the definite integral to find: (a) the value of (i) f(3) (ii) f(-1) (iii) 1'(6) (b)...
7. Evaluate the following integral. b) SO3(2-1)05 dz c) f (xl dx (15 marks) 8. A shaded region S is bounded by the curve y = the x-axis, the vertical lines x = 1 and x = V2. The x(4-2)1/4 region is rotated through 21 radians about x-axis to form a solid of revolution. Compute the exact volume of the solid of revolution formed. (10 marks) 9. Between the two towers of a suspension bridge that are 2 kilometres apart,...
1. (13 points) Use the limit of a Riemann Sum (i.e. sigma notation and the appropriate summation formulas) to evaluate the net-signed area between the graph of f(0) = 23 – 3 and the interval (0, 2). Let 27 be the right endpoint of the k-th subinterval (where all subintervals have equal width). Give your answer as a single integer or frac- tion, whichever is appropriate. Using any technique other than a limit of a Riemann Sum will earn no...
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