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Improper Integrals with Multiple Issues
Worksheet
Grade Level
Calculus II
CAL2.46 Improper Integrals with Multiple Issues
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For numerical work on proper subintervals, first write the improper integral \(\int_{0}
10
f(x)\,dx\), where \(f(x)=\frac{1}{\sqrt{x}(x-2)(x-5)\sqrt{10-x}}\), as a sum of limits. Which split is valid?
A
\(\lim_{u_1\to 2^-}\int_{0}
u_1
f(x)\,dx + \lim_{u_2\to 2^+}\int_{u_2}
3
f(x)\,dx + \lim_{u_3\to 5^-}\int_{3}
u_3
f(x)\,dx + \lim_{u_4\to 5^+}\int_{u_4}
7
f(x)\,dx + \lim_{u_5\to 10^-}\int_{7}
u_5
f(x)\,dx\)
B
\(\lim_{u_1\to 0^+}\int_{u_1}
1
f(x)\,dx + \lim_{u_2\to 2^-}\int_{1}
u_2
f(x)\,dx + \lim_{u_3\to 2^+}\int_{u_3}
3
f(x)\,dx + \lim_{u_4\to 5^-}\int_{3}
u_4
f(x)\,dx + \lim_{u_5\to 5^+}\int_{u_5}
7
f(x)\,dx + \lim_{u_6\to 10^-}\int_{7}
u_6
f(x)\,dx\)
C
\(\lim_{u_1\to 0^+}\int_{u_1}
1
f(x)\,dx + \lim_{u_2\to 2^-}\int_{1}
u_2
f(x)\,dx + \lim_{u_3\to 2^+}\int_{u_3}
3
f(x)\,dx + \lim_{u_4\to 5^-}\int_{3}
u_4
f(x)\,dx + \lim_{u_5\to 5^+}\int_{u_5}
10
f(x)\,dx\)
D
\(\int_{0}
2
f(x)\,dx + \int_{2}
10
f(x)\,dx\)
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