{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Midterm2_sample

# Midterm2_sample - and z = 1 x 2 y 2 1 4 Find a...

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

Past Exam Problems in Integrals Prof. Qiao Zhang Course 110.202 November 15, 2004 The following is a list of the problems concerning integrals that appeared in the midterm and ﬁnal exams of Calc III (110.202) within the last sev- eral years. You may use them to check your understanding of the relevant material. Some other exam problems may be found at http://reserves.library.jhu.edu/access/reserves/findit/exams/110/110202.php Note: These problems do not imply, in any sense, my taste or prefer- ence for our own exam. Some of the problems here may be more (or less) challenging than what will appear in our exam. 1. Evaluate the double integral ZZ D x 2 y 2 d x d y over the triangle D with vertices (0 , 0) , (1 , 0) , (1 , 2). 2. The shape of a platform is given by x 2 + y 2 (2 - z ) 2 , 0 z 1 . (a) Describe this shape in cylindrical coordinates. (b) Find the volume of this platform. 3. Find the volume of the solid enclosed by the two paraboloids z = 2( x 2 + y 2

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.

Unformatted text preview: ) and z = 1 + x 2 + y 2 . 1 4. Find a parametrization for the surface deﬁned by the intersection of the plane x + y + z = 1 with the cylinder x 2 + y 2 = 1. Use that parametrization to calculate the area of the surface. 5. Suppose that a particle follows the path r ( t ) = 2 cos(2 t ) i + 2 sin(2 t ) j + 3 t k . Then ﬁnd the total length of the path travelled by the particle from t = 0 to t = π/ 4. 6. Set up a double integral in polar coordinates to ﬁnd the volume of the solid which is bounded below by the paraboloid z = x 2 + y 2 and above by the plane z = 2 y . DO NOT EVALUATE! 7. Evaluate the integral Z 1 Z 1 y cos ± 1 2 πx 2 ² d x d y. 8. Find the volume of the region under the surface z = x 2 + 2 y and over the region in the ﬁrst quadrant between the line segment connecting (0 , 2) and (2 , 0) and the curve y = 4-x 2 . 2...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

Midterm2_sample - and z = 1 x 2 y 2 1 4 Find a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online