Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
The instructions and problems for exam 3 of a vector calculus course. The exam covers topics such as iterated integrals, regression lines, and double integrals to calculate volumes. Students are required to compute integrals, find regression lines, and determine volumes using given functions and regions.
Typology: Exams
1 / 7
This page cannot be seen from the preview
Don't miss anything!
Name:______________________________
Vector Calculus Exam 3 November 1 st
There are 6 problems and 144 points total. The point value of each question is indicated. Read each question carefully!
1. (24 points.) Compute the following iterated integrals
a)
(^2 3 2 ) 0 0
b)
1 1 0 0
x y
c)
2 2
1 1 0
r r
−
q d b p c a
2. (24 points.) Compute the regression line with the form a + bx for the points (-1,2), (0,-1), (1,1) using least squares and draw the regression line in the figure.
4. (24 points.) Compute the double integral of the function f ( , x y )= xy over the region
R = {( , x y ) : x^2 + y^2 ≤1}
5. (24 points.) Find the volume between the graph of the function f ( , x y ) = x y^2 and the xy − plane
over the region R = {( , x y ) : 0 ≤ x ≤ 1, − 1 ≤ y ≤0}
Extra credit Do not work on any of these until you have finished the rest of the exam!
A ) (6 points) Compute
(^1 ) 0
x y
−
B) (4 points) Notice in problems 2) and 3) that the mean of the x coordinates, x , and the mean of the
y coordinates, y , gives us a point ( , ) x y which lies on the regression line. This is a general fact which is closely related to the fact that the axis of rotation for a rigid body must pass through its center of mass. Use this fact to create a new set of three points whose regression line has the same y- intercept as the point set in problem 2)
C) (2 points) Is the volume in problem 5) larger or smaller than the volume in problem 6)?
