MIT18_014F10_pr_ex1_sols

# MIT18_014F10_pr_ex1_sols - Practice Exam 1 Solutions...

This preview shows pages 1–3. Sign up to view the full content.

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: Practice Exam 1- Solutions September 29, 2010 Problem 1: Compute 103 (2 x − 198) 2 √ x − 99 dx where here [ x ] is defined 99 to be the largest integer ≤ x . Solution By the properties of the integral, we know that the above is equal to 4 4 ( x ) 2 √ x dx. Now, [ √ x ] takes the value on [0 , 1) and the value 1 on [1 , 4) and the value 2 at x = 4. Thus, we can rewrite the integral 4 4 4 3 1 3 4 x 2 dx + 4 x 2 dx = 4 3 − 4 = 256 / 3 − 4 / 3 = 252 / 3 = 84 . 3 1 4 Problem 2: Let S be a square pyramid with base area r 2 and height h . Using Cavalieri’s Theorem, determine the volume of the pyramid. Solution Orient the square pyramid so that the base sits on the x − y plane and the top vertex sits on the z axis. Let a S ( h ) denote the cross-sectional area of S ∩ { z = h } , and note that this is a square that will be a function of r,h . To find the length of a side of the square at height h , we consider the line which contains the two points ( r, 0) and (0 ,h ). One form of the equation for this line is y = − r h x + h . Notice that here x is the side length of the square at...
View Full Document

## This note was uploaded on 01/18/2012 for the course MATH 18.014 taught by Professor Christinebreiner during the Fall '10 term at MIT.

### Page1 / 5

MIT18_014F10_pr_ex1_sols - Practice Exam 1 Solutions...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online