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Material Type: Notes; Class: UNIVERSITY CALCULUS I; Subject: MATHEMATICS; University: St. John's University-New York; Term: Fall 2008;
Typology: Study notes
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Worksheet (11/20/08) “Integration”
(1) Calculate the following integrals.
(a)
∫ (^2) x − 5 x^2 − 5 x dx
(b)
∫ (^2) x − 5 (x^2 − 5 x)^4 dx
(c)
x(1 + 2x) dx
(d)
0 e
x√ex (^) + 1 dx
(e)
3
√^3 x x − 2 dx^ (Use the substitution^ u^ =^ x^ −^ 2.)
(f)
x arctan x dx (Use integration by parts.)
(2) The rate of infection of a disease (in people per month) is given by the function I′(t) = (^) t^1002 + 1t, where t is the time in months since the disease broke out. Find the total number of infected people during the first four months of the disease.
(3) Suppose the temperature in a river at a point x in meters downstream from a factory that is discharging hot water into the river is given by T (x) = 400 − 0. 25 x^2. Find the average temperature over the interval [10, 40].
(4) A region R is enclosed by the curves y = x and y = x^2. Find the area of R.
(5) Does the improper integral
1
6 e−x^ dx converge or diverge? If it converges, what does it converge to?