# vctut03 - The University of Sydney School of Mathematics...

This preview shows pages 1–2. Sign up to view the full content.

The University of Sydney School of Mathematics and Statistics Tutorial 3 MATH2061: Vector Calculus Summer School 2012 1. Let I = integraldisplayintegraldisplay R xydxdy where R is the region in the xy plane bounded by y = x 2 , x = 0, y = 4 and where x 0. (a) Sketch the region R . (b) Evaluate I by integrating with respect to y first. (c) Evaluate I by integrating with respect to x first. 2. Sketch the region of integration and evaluate integraldisplay 1 0 integraldisplay 2 x x e x + y dydx. 3. Describe the region R , and use polar co-ordinates to evaluate the following: integraldisplayintegraldisplay R ( x 2 + y 2 ) 1 / 2 dA, R : x 2 + y 2 4 . 4. The sphere x 2 + y 2 + z 2 = 25 has a hole bored through it by the cylinder x 2 + y 2 = 4. Find the volume of the sphere that is removed. 5. Evaluate contintegraldisplay C ( x 2 xy 3 ) dx + ( y 2 2 xy ) dy, where C is the square with vertices at (0 , 0) , (2 , 0) , (2 , 2) , (0 , 2) and the contour is traversed in the anticlockwise direc- tion. 6. Evaluate contintegraldisplay C F · d r where F = (tan - 1 x + y 3 ) i (3 x 2 + sin y ) j , and C is the curve made up of the straight line segments from (0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern