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Section 5.8 Improper integrals (TI8)

Activity 5.8.1.

Consider \(\displaystyle f(x)=\frac{1}{x^2}\text{.}\) Compute the following definite integrals.

(a)

\(\displaystyle \int_{0.1}^1 \frac{1}{x^2} dx\)

(b)

\(\displaystyle \int_{0.01}^1 \frac{1}{x^2} dx\)

(c)

\(\displaystyle \int_{0.001}^1 \frac{1}{x^2} dx\)

Activity 5.8.2.

(a)

What do you notice about \(\displaystyle \int_{a}^1 \frac{1}{x^2} dx\) as \(a\) approached 0 in Activity 5.8.1?

  1. \(\displaystyle \int_{a}^1 \frac{1}{x^2} dx\) approaches a finite, non-zero constant.

  2. \(\displaystyle \int_{a}^1 \frac{1}{x^2} dx\) approaches \(0\text{.}\)

  3. \(\displaystyle \int_{a}^1 \frac{1}{x^2} dx\) approaches \(\infty\text{.}\)

  4. \(\displaystyle \int_{a}^1 \frac{1}{x^2} dx\) approaches \(-\infty\text{.}\)

  5. There is not enough information.

(b)

Why can't \(a=0\text{?}\)

Activity 5.8.3.

We still are considering \(f(x)=\frac{1}{x^2}\text{.}\) Compute the following definite integrals.

(a)

\(\displaystyle \int_{1}^{10} \frac{1}{x^2} dx\)

(b)

\(\displaystyle \int_{1}^{100} \frac{1}{x^2} dx\)

(c)

\(\displaystyle \int_{1}^{1000} \frac{1}{x^2} dx\)

Activity 5.8.4.

What do you notice about \(\displaystyle \int_{1}^a \frac{1}{x^2} dx\) as \(a\) approached \(\infty\) in Activity 5.8.3?

  1. \(\displaystyle \int_{1}^a \frac{1}{x^2} dx\) approaches a finite, non-zero constant.

  2. \(\displaystyle \int_{1}^a \frac{1}{x^2} dx\) approaches \(0\text{.}\)

  3. \(\displaystyle \int_{1}^a \frac{1}{x^2} dx\) approaches \(\infty\text{.}\)

  4. \(\displaystyle \int_{1}^a \frac{1}{x^2} dx\) approaches \(-\infty\text{.}\)

  5. There is not enough information.

Activity 5.8.5.

Compute the following definite integrals.

(a)

\(\displaystyle \int_{0.1}^1 \frac{1}{\sqrt{x}} dx\)

(b)

\(\displaystyle \int_{0.01}^1 \frac{1}{\sqrt{x}} dx\)

(c)

\(\displaystyle \int_{0.001}^1 \frac{1}{\sqrt{x}} dx\)

Activity 5.8.6.

(a)

What do you notice about the integral \(\displaystyle \int_{a}^1 \frac{1}{\sqrt{x}} dx\) as \(a\) approached 0 in Activity 5.8.5?

  1. \(\displaystyle \int_{a}^1 \frac{1}{\sqrt{x}} dx\) approaches a finite, non-zero constant.

  2. \(\displaystyle \int_{a}^1 \frac{1}{\sqrt{x}} dx\) approaches \(0\text{.}\)

  3. \(\displaystyle \int_{a}^1 \frac{1}{\sqrt{x}} dx\) approaches \(\infty\text{.}\)

  4. \(\displaystyle \int_{a}^1 \frac{1}{\sqrt{x}} dx\) approaches \(-\infty\text{.}\)

  5. There is not enough information.

Activity 5.8.7.

Compute the following definite integrals.

(a)

\(\displaystyle \int_{1}^{10} \frac{1}{\sqrt{x}} dx\)

(b)

\(\displaystyle \int_{1}^{100} \frac{1}{\sqrt{x}} dx\)

(c)

\(\displaystyle \int_{1}^{1000} \frac{1}{\sqrt{x}} dx\)

Activity 5.8.8.

(a)

What do you notice the integral \(\displaystyle \int_{1}^a \frac{1}{\sqrt{x}} dx\) as \(a\) approached \(\infty\) in Activity 5.8.7?

  1. \(\displaystyle \int_{1}^a \frac{1}{\sqrt{x}} dx\) approaches a finite, non-zero constant.

  2. \(\displaystyle \int_{1}^a \frac{1}{\sqrt{x}} dx\) approaches \(0\text{.}\)

  3. \(\displaystyle \int_{1}^a \frac{1}{\sqrt{x}} dx\) approaches \(\infty\text{.}\)

  4. \(\displaystyle \int_{1}^a \frac{1}{\sqrt{x}} dx\) approaches \(-\infty\text{.}\)

  5. There is not enough information.

Definition 5.8.9.

For a function \(f(x)\) and a constant \(c\text{,}\) we let \(\displaystyle \int_c^\infty f(x) dx\) denote

\begin{equation*} \int_c^\infty f(x) dx=\lim_{a\to\infty}\left( \int_c^a f(x) dx\right)\text{.} \end{equation*}

If this limit is a defined real number, then we say \(\displaystyle \int_c^\infty f(x) dx\) is convergent. Otherwise, it is divergent.

Similarly,

\begin{equation*} \int_{-\infty}^c f(x) dx=\lim_{a\to-\infty}\left( \int_a^c f(x) dx\right). \end{equation*}

Activity 5.8.10.

Which of the following is \(\displaystyle \int_1^a \frac{1}{x^2} dx\text{?}\)

  1. \(\displaystyle -\frac{1}{x}+C\)

  2. \(\displaystyle -\frac{1}{a}+C\)

  3. \(\displaystyle -\frac{1}{x}+1\)

  4. \(\displaystyle -\frac{1}{a}\)

  5. \(\displaystyle -\frac{1}{a}+1\)

Activity 5.8.11.

Using your choice in Activity 5.8.10, what is \(\lim_{a\to \infty} \int_1^a \frac{1}{x^2} dx\text{?}\)

  1. \(\displaystyle 0\)

  2. \(\displaystyle 1\)

  3. \(\displaystyle 2\)

  4. \(\displaystyle \infty\)

  5. \(\displaystyle -\infty\)

Activity 5.8.12.

(a)

Find \(\displaystyle \int_{\pi/2}^a \cos(x)dx\text{.}\)

(b)

Which of the following is true about \(\int_{\pi/2}^\infty \cos(x)dx\text{?}\)

  1. \(\displaystyle \int_{\pi/2}^\infty \cos(x)dx\) is convergent.

  2. \(\displaystyle \int_{\pi/2}^\infty \cos(x)dx\) is divergent.

  3. More information is needed.

Definition 5.8.13.

For a function \(f(x)\) with a vertical asymptote at \(x=b\) and a constant \(c\text{,}\) we let \(\displaystyle \int_c^b f(x) dx\) denote

\begin{equation*} \int_c^b f(x) dx=\lim_{a\to b^{-}}\left( \int_c^a f(x) dx\right)\text{.} \end{equation*}

Similarly,

\begin{equation*} \int_{b}^c f(x) dx=\lim_{a\to b^{+}}\left( \int_a^c f(x) dx\right)\text{.} \end{equation*}

Activity 5.8.14.

(a)

Find \(\displaystyle \int_{a}^1 \frac{1}{\sqrt{x}} dx\text{.}\)

(b)

Using the above result and an appropriate limit, find \(\displaystyle \int_{0}^1 \frac{1}{\sqrt{x}} dx\text{.}\)

Activity 5.8.15.

(a)

Find \(\displaystyle \int_{a}^1 \frac{1}{x^2} dx\text{.}\)

(b)

Using the above result and an appropriate limit, find \(\displaystyle \int_{0}^1 \frac{1}{x^2} dx\text{.}\)

Activity 5.8.17.

(a)

If \(0< p< 1\text{,}\) which of the following statements must be true? Select all that apply.

  1. \(\displaystyle 1-p< 0\)

  2. \(\displaystyle 1-p> 0\)

  3. \(\displaystyle 1-p< 1\)

  4. \(\displaystyle \int_1^\infty \frac{1}{x^p} dx\) converges.

  5. \(\displaystyle \int_1^\infty \frac{1}{x^p} dx\) diverges.

(b)

If \(p>1\text{,}\) which of the following statements must be true? Select all that apply.

  1. \(\displaystyle 1-p< 0\)

  2. \(\displaystyle 1-p> 0\)

  3. \(\displaystyle 1-p< 1\)

  4. \(\displaystyle \int_1^\infty \frac{1}{x^p} dx\) converges.

  5. \(\displaystyle \int_1^\infty \frac{1}{x^p} dx\) diverges.

Activity 5.8.18.

(a)

If \(0< p< 1\text{,}\) which of the following statements must be true?

  1. \(\displaystyle \int_0^1 \frac{1}{x^p} dx\) converges.

  2. \(\displaystyle \int_0^1 \frac{1}{x^p} dx\) diverges.

(b)

If \(p>1\text{,}\) which of the following statements must be true?

  1. \(\displaystyle \int_0^1 \frac{1}{x^p} dx\) converges.

  2. \(\displaystyle \int_0^1 \frac{1}{x^p} dx\) diverges.

Activity 5.8.19.

Consider when \(p=1\text{.}\) Then \(\frac{1}{x^p}=\frac{1}{x}\) and \(\displaystyle \int \frac{1}{x^p}\ dx=\displaystyle \int \frac{1}{x}\ dx=\ln|x|+C\text{.}\)

(a)

What can we conclude about \(\displaystyle \int_1^\infty \frac{1}{x} dx\text{?}\)

  1. \(\displaystyle \int_1^\infty \frac{1}{x} dx\) converges.

  2. \(\displaystyle \int_1^\infty \frac{1}{x} dx\) diverges.

  3. There is not enough information to determine whether this integral converges or diverges.

(b)

What can we conclude about \(\displaystyle \int_0^1 \frac{1}{x} dx\text{?}\)

  1. \(\displaystyle \int_0^1 \frac{1}{x} dx\) converges.

  2. \(\displaystyle \int_0^1 \frac{1}{x} dx\) diverges.

  3. There is not enough information to determine whether this integral converges or diverges.

Activity 5.8.21.

Consider the plots of \(f(x), g(x), h(x) \) where \(0 < g(x) < f(x) < h(x)\text{.}\)

Plots of positive functions f(x), g(x) where f(x) is an upper bound of g(x).
Figure 82. Plots of \(f(x), g(x), h(x)\)

If \(\displaystyle \int_1^\infty f(x) dx\) is convergent, what can we say about \(g(x), h(x)\text{?}\)

  1. \(\displaystyle \int_1^\infty g(x) dx\) and \(\displaystyle \int_1^\infty h(x) dx\) are both convergent.

  2. \(\displaystyle \int_1^\infty g(x) dx\) and \(\displaystyle \int_1^\infty h(x) dx\) are both divergent.

  3. Whether or not \(\displaystyle \int_1^\infty g(x) dx\) and \(\displaystyle \int_1^\infty h(x) dx\) are convergent or divergent cannot be determined.

  4. \(\displaystyle \int_1^\infty g(x) dx\) is convergent and \(\displaystyle \int_1^\infty h(x) dx\) is divergent.

  5. \(\displaystyle \int_1^\infty g(x) dx\) is convergent and \(\displaystyle \int_1^\infty h(x) dx\) could be either convergent or divergent.

Activity 5.8.22.

Consider the plots of \(f(x), g(x), h(x) \) where \(0 < g(x) < f(x) < h(x)\text{.}\)

Plots of positive functions f(x), g(x) where f(x) is an upper bound of g(x).
Figure 83. Plots of \(f(x), g(x), h(x)\)
If \(\displaystyle \int_1^\infty f(x) dx\) is divergent, what can we say about \(g(x), h(x)\text{?}\)

  1. \(\displaystyle \int_1^\infty g(x) dx\) and \(\displaystyle \int_1^\infty h(x) dx\) are both convergent.

  2. \(\displaystyle \int_1^\infty g(x) dx\) and \(\displaystyle \int_1^\infty h(x) dx\) are both divergent.

  3. Whether or not \(\displaystyle \int_1^\infty g(x) dx\) and \(\displaystyle \int_1^\infty h(x) dx\) are convergent or divergent cannot be determined.

  4. \(\displaystyle \int_1^\infty g(x) dx\) could be either convergent or dicergent and \(\displaystyle \int_1^\infty h(x) dx\) is divergent.

  5. \(\displaystyle \int_1^\infty g(x) dx\) is convergent and \(\displaystyle \int_1^\infty h(x) dx\) is divergent.

Activity 5.8.23.

Consider the plots of \(f(x), g(x), h(x) \) where \(0 < g(x) < f(x) < h(x)\text{.}\)

Plots of positive functions f(x), g(x) and h(x).
Figure 84. Plots of \(f(x), g(x), h(x)\)
If \(\displaystyle \int_0^1 f(x) dx\) is convergent, what can we say about \(g(x)\) and \(h(x)\text{?}\)

  1. \(\displaystyle \int_0^1 g(x) dx\) and \(\displaystyle \int_0^1 h(x) dx\) are both convergent.

  2. \(\displaystyle \int_0^1 g(x) dx\) and \(\displaystyle \int_0^1 h(x) dx\) are both divergent.

  3. Whether or not \(\displaystyle \int_0^1 g(x) dx\) and \(\displaystyle \int_0^1 h(x) dx\) are convergent or divergent cannot be determined.

  4. \(\displaystyle \int_0^1 g(x) dx\) is convergent and \(\displaystyle \int_0^1 h(x) dx\) is divergent.

  5. \(\displaystyle \int_0^1 g(x) dx\) is convergent and \(\displaystyle \int_0^1 h(x) dx\) can either be convergent or divergent.

Activity 5.8.24.

Consider the plots of \(f(x), g(x), h(x) \) where \(0 < g(x) < f(x) < h(x)\text{.}\)

Plots of positive functions f(x), g(x) and h(x).
Figure 85. Plots of \(f(x), g(x), h(x)\)
If \(\displaystyle \int_0^1 f(x) dx\) is dinvergent, what can we say about \(g(x)\) and \(h(x)\text{?}\)

  1. \(\displaystyle \int_0^1 g(x) dx\) and \(\displaystyle \int_0^1 h(x) dx\) are both convergent.

  2. \(\displaystyle \int_0^1 g(x) dx\) and \(\displaystyle \int_0^1 h(x) dx\) are both divergent.

  3. Whether or not \(\displaystyle \int_0^1 g(x) dx\) and \(\displaystyle \int_0^1 h(x) dx\) are convergent or divergent cannot be determined.

  4. \(\displaystyle \int_0^1 g(x) dx\) can be either convergent or divergent and \(\displaystyle \int_0^1 h(x) dx\) is divergent.

  5. \(\displaystyle \int_0^1 g(x) dx\) is convergent and \(\displaystyle \int_0^1 h(x) dx\) is divergent.

Activity 5.8.26.

Compare \(\frac{1}{x^3+1}\) to one of the following functions where \(x>2\) and use this to determine if \(\displaystyle \int_2^\infty \frac{1}{x^3+1}dx\) is convergent or divergent.

  1. \(\displaystyle \displaystyle\frac{1}{x}\)

  2. \(\displaystyle \displaystyle\frac{1}{\sqrt{x}}\)

  3. \(\displaystyle \displaystyle\frac{1}{x^2}\)

  4. \(\displaystyle \displaystyle\frac{1}{x^3}\)

Subsection 5.8.1 Videos

Figure 86. Video: I can compute improper integrals, \(p>1\)
Figure 87. Video: I can compute improper integrals, \(p < 1\)