mathsTutorial 8.pdf

MATH F111 Tutorial 8 2 xyexy dA, over...

MATH F111 Tutorial 8 2 xyexy dA, over RR 1. (Evaluating Double Integrals over Rectangles) Evaluate the double integral R the region R : 0 ≤ x ≤ 2, 0 ≤ y ≤ 1. 2. Find the volume of the region bounded above by the surface z = 2 sin x cos y and below by the rectangle R : 0 ≤ x ≤ π/2, 0 ≤ y ≤ π/4. 3. Use Fubini’s theorem to evaluate the following integral. Z 1Z 3 xexy dx dy 0 0 4. Sketch the region of integration : 0 ≤ y ≤ 1, 0 ≤ x ≤ sin−1 y. 5. Sketch the region of integration and evaluate the integral Z π Z sin x y dy dx. 0 0 6. The following expression gives the integral over a region in a Cartesian coordinate plane (the tu-plane). Sketch the region and evaluate the integral. Z π/3 Z sec t 3 cos t du dt. −π/3 0 7. (Integrals over Unbounded Regions) Evaluate the following improper integral as iterated integral: Z ∞Z ∞ 1 2 + 1)(y 2 + 1) dx dy. −∞ −∞ (x 8. (Noncircular cylinder) A solid right (noncircular) cylinder has its base R in the xy-plane and is bounded above by the paraboloid z = x2 + y 2. The cylinder’s volume is Z 1Z y Z 2 Z 2−y V = (x2 + y 2 ) dx dy + (x2 + y 2 ) dx dy 0 0 1 0 Sketch the base region R and express the cylinder’s volume as a single iterated integral with the order of integration reversed. Then evaluate the integral to find the volume. 9. How would you evaluate the double integral of a continuous function f (x, y) over the region R in the xy-plane enclosed by the triangle with vertices (0, 1), (2, 0), and (1, 2)? Give reasons for your answer. 1 10. Sketch the region bounded by the curves y = ln x and y = 2 ln x and the line x = e, in the first quadrant. Then express the region’s area as an iterated double integral and evaluate the integral. 11. Sketch the region bounded by the parabolas x = y 2 − 1 and x = 2y 2 − 2. Then express the region’s area as an iterated double integral and evaluate the integral. 12. (Geometric area) Find the area of the circular washer with outer radius 2 and inner radius 1, using (a) Fubini’s theorem, and (b) simple geometry. 2