BackImproper Integrals quiz #1
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BackHow do you evaluate the improper integral ∫₀^∞ e^x dx, and does it converge or diverge?
To evaluate ∫₀^∞ e^x dx, rewrite it as a limit: limₜ→∞ ∫₀^t e^x dx. Integrating gives e^x from 0 to t, so the result is limₜ→∞ (e^t - 1). As t approaches infinity, e^t grows without bound, so the integral diverges. What is the correct procedure for evaluating an improper integral with both bounds infinite, such as ∫_{-∞}^{∞} f(x) dx?
To evaluate ∫_{-∞}^{∞} f(x) dx, split it into two integrals at a constant c: ∫_{-∞}^c f(x) dx + ∫_c^{∞} f(x) dx. Then, use limits for each part: limₜ→-∞ ∫ₜ^c f(x) dx and limₛ→∞ ∫_c^s f(x) dx. The original integral converges only if both limits exist and are finite. What is an improper integral?
An improper integral is an integral where one or both of the bounds are infinite, or the integrand becomes infinite within the interval. How do you rewrite an integral with an upper bound of infinity, such as ∫ₐ^∞ f(x) dx?
You rewrite it as a limit: limₜ→∞ ∫ₐ^t f(x) dx. How do you handle an integral with a lower bound of negative infinity, such as ∫_{-∞}^b f(x) dx?
You rewrite it as limₜ→-∞ ∫ₜ^b f(x) dx. What is the procedure for evaluating an improper integral with both bounds infinite, like ∫_{-∞}^{∞} f(x) dx?
Split it at a constant c: ∫_{-∞}^c f(x) dx + ∫_c^{∞} f(x) dx, and use limits for each part. What does it mean for an improper integral to converge?
It means the limit exists and the integral evaluates to a finite number. What does it mean for an improper integral to diverge?
It means the limit does not exist or the integral grows without bound. Evaluate ∫₀^∞ e^x dx and state if it converges or diverges.
It diverges because limₜ→∞ (e^t - 1) grows without bound. Evaluate ∫_{-∞}^0 e^x dx and state if it converges or diverges.
It converges to 1 because limₜ→-∞ (1 - e^t) approaches 1 as e^t goes to 0.
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