Neetesh Kumar | December 10, 2024
Calculus Homework Help
This is the solution to Math 132
Assignment: 7.8 Question Number 1
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Step-by-step solution:
(a) Improper Integral Explanation
The given integral is:
∫23x−2xdx
- The denominator x−2 becomes 0 at x=2, which is the lower limit of integration.
- This creates an infinite discontinuity within the range of integration.
Reason:
Since the integral has an infinite discontinuity at the lower limit x=2, it is classified as a Type 2 improper integral.
(b) Improper Integral Explanation
The given integral is:
∫0∞1+x31dx
- The upper limit of integration is x=∞, which makes the interval of integration infinite.
Reason:
Since the integral has an infinite interval of integration, it is classified as a Type 1 improper integral.
(c) Improper Integral Explanation
The given integral is:
∫−∞∞x2e−x2dx
- The limits of integration are −∞ and ∞, which means the interval of integration is infinite.
Reason:
Since the integral has an infinite interval of integration, it is classified as a Type 1 improper integral.
(d) Improper Integral Explanation
The given integral is:
∫0π/4cot(x)dx
- The integrand cot(x) has a discontinuity at x=0, which is the lower limit of integration.
Reason:
Since the integral has an infinite discontinuity at the lower limit x=0, it is classified as a Type 2 improper integral.
Final Classification Summary:
(a): Type 2 improper integral (infinite discontinuity at x=2)
(b): Type 1 improper integral (infinite interval of integration)
(c): Type 1 improper integral (infinite interval of integration)
(d): Type 2 improper integral (infinite discontinuity at x=0)
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