6.E: Using Definite Integrals (Exercises)

1. Area between two power functions

2. Area between two trigonometric functions

3. Area between two curves

4. Arc length of a curve

5

Find the exact area of each described region.

1. The finite region between the curves $x = y(y-2)$ and $x=-(y-1)(y-3)\text{.}$
2. The region between the sine and cosine functions on the interval $[\frac{\pi}{4}, \frac{3\pi}{4}]\text{.}$
3. The finite region between $x = y^2 - y - 2$ and $y = 2x-1\text{.}$
4. The finite region between $y = mx$ and $y = x^2-1\text{,}$ where $m$ is a positive constant.

6

Let $f(x) = 1-x^2$ and $g(x) = ax^2 - a\text{,}$ where $a$ is an unknown positive real number. For what value(s) of $a$ is the area between the curves $f$ and $g$ equal to 2?

7

Let $f(x) = 2-x^2\text{.}$ Recall that the average value of any continuous function $f$ on an interval $[a,b]$ is given by $\frac{1}{b-a} \int_a^b f(x) \, dx\text{.}$

1. Find the average value of $f(x) = 2-x^2$ on the interval $[0,\sqrt{2}]\text{.}$ Call this value $r\text{.}$
2. Sketch a graph of $y = f(x)$ and $y = r\text{.}$ Find their intersection point(s).
3. Show that on the interval $[0,\sqrt{2}]\text{,}$ the amount of area that lies below $y = f(x)$ and above $y = r$ is equal to the amount of area that lies below $y = r$ and above $y = f(x)\text{.}$
4. Will the result of (c) be true for any continuous function and its average value on any interval? Why?

1. Solid of revolution from one function about the $x$-axis

2. Solid of revolution from one function about the $y$-axis

3. Solid of revolution from two functions about the $x$-axis

4. Solid of revolution from two functions about a horizontal line

5. Solid of revolution from two functions about a different horizontal line

6. Solid of revolution from two functions about a vertical line

7

Consider the curve $f(x) = 3 \cos(\frac{x^3}{4})$ and the portion of its graph that lies in the first quadrant between the $y$-axis and the first positive value of $x$ for which $f(x) = 0\text{.}$ Let $R$ denote the region bounded by this portion of $f\text{,}$ the $x$-axis, and the $y$-axis.

1. Set up a definite integral whose value is the exact arc length of $f$ that lies along the upper boundary of $R\text{.}$ Use technology appropriately to evaluate the integral you find.
2. Set up a definite integral whose value is the exact area of $R\text{.}$ Use technology appropriately to evaluate the integral you find.
3. Suppose that the region $R$ is revolved around the $x$-axis. Set up a definite integral whose value is the exact volume of the solid of revolution that is generated. Use technology appropriately to evaluate the integral you find.
4. Suppose instead that $R$ is revolved around the $y$-axis. If possible, set up an integral expression whose value is the exact volume of the solid of revolution and evaluate the integral using appropriate technology. If not possible, explain why.

8

Consider the curves given by $y = \sin(x)$ and $y = \cos(x)\text{.}$ For each of the following problems, you should include a sketch of the region/solid being considered, as well as a labeled representative slice.

1. Sketch the region $R$ bounded by the $y$-axis and the curves $y = \sin(x)$ and $y = \cos(x)$ up to the first positive value of $x$ at which they intersect. What is the exact intersection point of the curves?
2. Set up a definite integral whose value is the exact area of $R\text{.}$
3. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving $R$ about the $x$-axis.
4. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving $R$ about the $y$-axis.
5. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving $R$ about the line $y = 2\text{.}$
6. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving $R$ about the line $x = -1\text{.}$

9

Consider the finite region $R$ that is bounded by the curves $y = 1+\frac{1}{2}(x-2)^2\text{,}$ $y=\frac{1}{2}x^2\text{,}$ and $x = 0\text{.}$

1. Determine a definite integral whose value is the area of the region enclosed by the two curves.
2. Find an expression involving one or more definite integrals whose value is the volume of the solid of revolution generated by revolving the region $R$ about the line $y = -1\text{.}$
3. Determine an expression involving one or more definite integrals whose value is the volume of the solid of revolution generated by revolving the region $R$ about the $y$-axis.
4. Find an expression involving one or more definite integrals whose value is the perimeter of the region $R\text{.}$

1. Center of mass for a linear density function

2. Center of mass for a nonlinear density function

3. Interpreting the density of cars on a road

4. Center of mass in a point-mass system

5

Let a thin rod of length $a$ have density distribution function $\rho(x) = 10e^{-0.1x}\text{,}$ where $x$ is measured in cm and $\rho$ in grams per centimeter.

1. If the mass of the rod is 30 g, what is the value of $a\text{?}$
2. For the 30g rod, will the center of mass lie at its midpoint, to the left of the midpoint, or to the right of the midpoint? Why?
3. For the 30g rod, find the center of mass, and compare your prediction in (b).
4. At what value of $x$ should the 30g rod be cut in order to form two pieces of equal mass?

6

Consider two thin bars of constant cross-sectional area, each of length 10 cm, with respective mass density functions $\rho(x) = \frac{1}{1+x^2}$ and $p(x) = e^{-0.1x}\text{.}$

1. Find the mass of each bar.
2. Find the center of mass of each bar.
3. Now consider a new 10 cm bar whose mass density function is $f(x) = \rho(x) + p(x)\text{.}$
   1. Explain how you can easily find the mass of this new bar with little to no additional work.
   2. Similarly, compute $\int_0^{10} xf(x) \, dx$ as simply as possible, in light of earlier computations.
   3. True or false: the center of mass of this new bar is the average of the centers of mass of the two earlier bars. Write at least one sentence to say why your conclusion makes sense.

7

Consider the curve given by $y = f(x) = 2xe^{-1.25x} + (30-x) e^{-0.25(30-x)}\text{.}$

1. Plot this curve in the window $x = 0 \ldots 30\text{,}$ $y = 0 \ldots 3$ (with constrained scaling so the units on the $x$ and $y$ axis are equal), and use it to generate a solid of revolution about the $x$-axis. Explain why this curve could generate a reasonable model of a baseball bat.
2. Let $x$ and $y$ be measured in inches. Find the total volume of the baseball bat generated by revolving the given curve about the $x$-axis. Include units on your answer.
3. Suppose that the baseball bat has constant weight density, and that the weight density is $0.6$ ounces per cubic inch. Find the total weight of the bat whose volume you found in (b).
4. Because the baseball bat does not have constant cross-sectional area, we see that the amount of weight concentrated at a location $x$ along the bat is determined by the volume of a slice at location $x\text{.}$ Explain why we can think about the function $\rho(x) = 0.6 \pi f(x)^2$ (where $f$ is the function given at the start of the problem) as being the weight density function for how the weight of the baseball bat is distributed from $x = 0$ to $x = 30\text{.}$
5. Compute the center of mass of the baseball bat.

1. An improper integral on a finite interval

2. An improper integral on an infinite interval

3. An improper integral involving a ratio of exponential functions

4. A subtle improper integral

5. An improper integral involving a ratio of trigonometric functions

6

Determine, with justification, whether each of the following improper integrals converges or diverges.

1. $\displaystyle \int_e^{\infty} \frac{\ln(x)}{x} \, dx$
2. $\displaystyle \int_e^{\infty} \frac{1}{x\ln(x)} \, dx$
3. $\displaystyle \int_e^{\infty} \frac{1}{x(\ln(x))^2} \, dx$
4. $\int_e^{\infty} \frac{1}{x(\ln(x))^p} \, dx\text{,}$ where $p$ is a positive real number
5. $\displaystyle \int_0^1 \frac{\ln(x)}{x} \, dx$
6. $\displaystyle \int_0^1 \ln(x) \, dx$

7

Sometimes we may encounter an improper integral for which we cannot easily evaluate the limit of the corresponding proper integrals. For instance, consider $\int_1^{\infty} \frac{1}{1+x^3} \, dx\text{.}$ While it is hard (or perhaps impossible) to find an antiderivative for $\frac{1}{1+x^3}\text{,}$ we can still determine whether or not the improper integral converges or diverges by comparison to a simpler one. Observe that for all $x \gt 0\text{,}$ $1 + x^3 \gt x^3\text{,}$ and therefore

It therefore follows that

for every $b \gt 1\text{.}$ If we let $b \to \infty$ so as to consider the two improper integrals $\int_1^\infty \frac{1}{1+x^3} \, dx$ and $\int_1^\infty \frac{1}{x^3} \, dx\text{,}$ we know that the larger of the two improper integrals converges. And thus, since the smaller one lies below a convergent integral, it follows that the smaller one must converge, too. In particular, $\int_1^\infty \frac{1}{1+x^3} \, dx$ must converge, even though we never explicitly evaluated the corresponding limit of proper integrals. We use this idea and similar ones in the exercises that follow.

1. Explain why $x^2 + x + 1 \gt x^2$ for all $x \ge 1\text{,}$ and hence show that $\int_1^{\infty} \frac{1}{x^2 + x + 1} \, dx$ converges by comparison to $\int_1^{\infty} \frac{1}{x^2} \, dx\text{.}$
2. Observe that for each $x \gt 1\text{,}$ $\ln(x) \lt x\text{.}$ Explain why
   $$\int_2^b \frac{1}{x} \, dx \lt \int_2^b \frac{1}{\ln(x)} \,dx$$

   for each $b \gt 2\text{.}$ Why must it be true that $\int_2^b \frac{1}{\ln(x)} \, dx$ diverges?

3. Explain why $\sqrt{\frac{x^4+1}{x^4}} \gt 1$ for all $x \gt 1\text{.}$ Then, determine whether or not the improper integral
   $$\int_1^{\infty} \frac{1}{x} \cdot \sqrt{\frac{x^4+1}{x^4}} \, dx$$

   converges or diverges.