MAT 271 Integral Mastery Test #1: Solutions
(1) Transform the integral ln x ex dx into another integral using integration by parts. You do not need to evaluate this second integral.
Solution: Let u = ln x and v = ex . Then u =
1 and v = ex , so x 1
MAT 271 Integral Mastery Test #2: Solutions
1
(1) Rewrite the integral
0
x2 sin1 x dx in terms of another integral using integration by
parts. You do not need to evaluate this second integral.
Solution: Use integration by parts, with u = sin1 x a
MAT 271 Maple Project: Approximating
The goal of this project is to approximate the number to thirty decimal places using two methods: approximate integration and approximation by series. You will do this in several steps. The numbered steps are
Solutions to MAT 271 Test #1
(1) (15 points) Evaluate the integral t2 sin(2t) dt. Solution: Use integration by parts twice: cos(2t) 2 2t cos(2t) dt 2 t cos(2t) dt sin(2t) dt 2 u=t u =1 v = cos(2t) u = t2 u = 2t v = sin(2t)
t2 sin(2t) dt =
MATH 271 Test #3T Solutions
(1) (20 points) A tank in the shape of a sphere with a radius of 5 meters is lled halfway with water (density: 1000 kg/m3 ). The center of the tank is 20 meters underground. How much work does it take to pump the water to
MATH 271 Test #4T Solutions
You do not need to evaluate the integrals in problems (2)(5); just set them up.
(1) (20 points) Consider the curve which is represented by the parametric equations x = 1 + t + t2 , y = 1 + et , where t is any real number.
MAT 271 Test #5 Solutions
(1) (20 points) Find the radius of convergence for the series
k=5
(x 2)k . You do not k 2k
need to nd the interval of convergence. Solution: To nd the radius of convergence for a power series, you use the ratio (x 2)k
MAT 271 Topics
The following is a list of topics from MAT 271, which show you what you need to know for the nal exam, and the sections of the book which deal with these topics. Items marked with an asterisk () are especially important. Know when an
Based on the graph, the three groups of organisms are roughly the same proportion. The vascular plants seem to always have the highest number of species in each category except for a 9 specie difference in the possibly extinct category. The invertebr