Applications of Integration

find an equation that gives y as an implicit function of x. Then draw the continuous curve that satisfies this differential equation and passes through the point (2, 0).

$\frac{dy }{dx}=\frac{-x}{y}$

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)={(2x+3)}^{3/2}$, $\left[a,b\right]=\left[-1,1\right]$ 

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral. 

$f\left(x\right)=2{(1-x)}^{3/2}+3$, $\left[a,b\right]=\left[-2,0\right]$

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=\sqrt{9-x^2}$, $\left[a,b\right]=\left[-3,3\right]$ 

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=\sqrt{1-x^2}$, $\left[a,b\right]=\left[-1,1\right]$ 

Consider the region between f(x) = x − 2 and the x-axis on [2, 5]. For each line of rotation given in Exercises 33–38, use the shell method to construct definite integrals to find the volume of the resulting solid 

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=\frac{1}{3}{x}^{3/2}-x^{1/2}$, $\left[a,b\right]=\left[0,1\right]$ 

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)={(1-x^{2/3})}^{3/2}$, $\left[a,b\right]=\left[0,1\right]$ 

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)={(4-x^{2/3})}^{3/2}$, $\left[a,b\right]=\left[0,2\right]$. 

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=x^2$, $\left[a,b\right]=\left[-1,1\right]$ 

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=x^2-\frac{1}{8}\ln x$, $\left[a,b\right]=\left[1,2\right]$ 

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=\ln \left(\cos x\right)$, $\left[a,b\right]=\left[0,\frac{\pi }{4}\right]$

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=\ln \left(\sin x\right)$,  $\left[a,b\right]=\left[\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4}\right]$

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=\frac{x^4+3}{6x}$, $\left[a,b\right]=\left[1,3\right]$

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.

${\int }_{-2}^5\sqrt{1+9e^{6x}}dx$ 

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.

${\int }_0^{\mathrm{\pi }}\sqrt{1+{\sin }^{2}x}dx$ 

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.

${\int }_2^3\sqrt{\frac{x^4+1}{x^4}}dx$ 

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.

${\int }_0^2\sqrt{1+36x}dx$

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.

${\int }_0^{\mathrm{\pi }/4}secxdx$ 

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.

${\int }_0^1\sqrt{1+9x^4}dx$ 

You may have noticed that even very simple functions give rise to arc length integrals that we have no idea how to compute. Use a graphing calculator to approximate a definite integral that represents the arc length of the given function f(x) on the interval [a, b]. 

$f\left(x\right)=x^3$, $\left[a,b\right]=\left[-1,1\right]$

Use definite integrals to find the volume of each solid of revolution described in Exercises $49-61$. (It is your choice whether to use disks/washers or shells in these exercises.)

The region bounded the graph of $f\left(x\right)=e^x$and the line $y=e$on $\left[0,1\right],$ revolved around the x-axis.

You may have noticed that even very simple functions give rise to arc length integrals that we have no idea how to compute. Use a graphing calculator to approximate a definite integral that represents the arc length of the given function f(x) on the interval [a, b]. 

$f\left(x\right)=x^2+1$, $\left[a,b\right]=\left[0,4\right]$

You may have noticed that even very simple functions give rise to arc length integrals that we have no idea how to compute. Use a graphing calculator to approximate a definite integral that represents the arc length of the given function f(x) on the interval [a, b].

$f\left(x\right)=3x^2-1$, $\left[a,b\right]=\left[1,2\right]$ 

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Every sum is a Riemann sum and can be turned into a definite integral.

(b) True or False: Every sum involving only continuous functions is a Riemann sum and can be turned into a definite integral.

(c) True or False: The volume of a disk can be obtained by multiplying its thickness by the circumference of a circle of the same radius.

(d) True or False: The volume of a disk can be obtained by multiplying its thickness by the area of a circle of the same radius.

(e) True or False: The volume of a cylinder can be obtained by multiplying the height of the cylinder by the area of a circle of the same radius.

(f) True or False: The volume of a washer can be expressed as the difference of the volume of two disks.

(g) True or False: The volume of a right cone is exactly one third of the volume of a cylinder with the same radius and height.

(h) True or False: The volume of a sphere of radius r is $V=\frac{4}{3}{\mathrm{\pi r}}^3$

Calculate each of the following definite integrals, using integration techniques and fundamental theorem of calculus

$$\begin{gathered} {\int }_{-3}^3(9-x^2)dx \\ {\int }_0^1(x^4+2x^2)dx \\ {\int }_0^1\left(4\right)dx \\ {\int }_{-3}^3(4-9{\left(\ln x\right)}^2)dx \\ {\int }_0^2(4y^2-y^4)dy \\ {\int }_0^1(2-\sqrt{y})dy \end{gathered}$$

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A region that, when revolved around the x-axis, has both disk and washer cross sections.

(b) A region that, when revolved around the y-axis, has both disk and washer cross sections.

(c) A solid of revolution for which it is not possible to use
the disk or washer method.

Consider the rectangle bounded by y = 3 and y = 0 on the x-interval $[2,2.5]$.

(a) What is the volume of the disk obtained by rotating this rectangle around the x-axis?

(b) What is the volume of the washer obtained by rotating this rectangle around the line y=5?

Consider the region between $f\left(x\right)=5-x^2$ and the $x$-axis between $x=0$ and $x=4$. Draw a Riemann sum approximation of the area of this region, using a midpoint sum with four rectangles, and explain how it is related to a four-disk approximation of the solid obtained by rotating the region around the$x$-axis.

For a four-disk approximation of the volume of the solid obtained from the region between$f\left(x\right)=5-x^2$ and the$x$-axis between $x=0$ and $x=4$ by rotating around the $x$-axis, illustrate and calculate

(a) $\Delta x$ and each $x_k;$

(b) some $x_k^{*}$ in each subinterval $\left[x_{k-1},x_k\right]$;

(c) each $f\left(x_k^{*}\right)$;

(d) the volume of the second disk.

For a four-washer approximation of the volume of the solid obtained from the region between $f\left(x\right)=x^2$ and the$y$-axis between $y=0$ and $y=4$ by rotating around the $x$-axis, illustrate and calculate

(a) $\Delta x$and each $x_{k;}$

(b) some$x_k^{*}$ in each subinterval $\left[x_{k-1},x_k\right]$;

(c) each $f\left(x_k^{*}\right)$;

(d) the volume of the second washer.

For a four-disk approximation of the volume of the solid obtained from the region between $f\left(x\right)=x^2$ and the $y$ axis between $y=0$ and $y=4$by rotating around the $y$-axis, illustrate and calculate

(a) $\Delta y$ and each $y_{ki}$

(b) some $y_k^{*}$ in each subinterval$\left[y_{k-1},y_k\right]$;

(c) each $f^{-1}\left(y_k^{*}\right)$;

(d) the volume of the second disk.

Write each of the limits in Exercises 9-11 in terms of definite integrals, and identify a solid of revolution whose volume is represented by that definite integral.

$$\begin{gathered} \mathrm{Write} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{limits} \mathrm{in} \mathrm{Exercises} 9–11 \mathrm{in} \mathrm{terms} \mathrm{of} \mathrm{definite} \mathrm{integrals}, \mathrm{and} \mathrm{identify} \\ \mathrm{a} \mathrm{solid} \mathrm{of} \mathrm{revolution} \mathrm{whose} \mathrm{volume} \mathrm{is} \mathrm{represented} \mathrm{by} \mathrm{that} \mathrm{definite} \mathrm{integral}: \\ \underset{\mathrm{n}\to \infty }{\lim } \sum _{\mathrm{k}=1}^{\mathrm{n}}\mathrm{\pi }{(1+{\mathrm{y}}_{\mathrm{k}}^{*})}^2 \left(\frac{3}{\mathrm{n}}\right), \mathrm{with} {\mathrm{y}}_{\mathrm{k}}^{*}={\mathrm{y}}_{\mathrm{k}}=1+\mathrm{k}\left(\frac{3}{\mathrm{n}}\right) \end{gathered}$$

$$\begin{gathered} \mathrm{Write} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{limits} \mathrm{in} \mathrm{Exercises} 9–11 \mathrm{in} \mathrm{terms} \mathrm{of} \mathrm{definite} \mathrm{integrals}, \mathrm{and} \mathrm{identify} \\ \mathrm{a} \mathrm{solid} \mathrm{of} \mathrm{revolution} \mathrm{whose} \mathrm{volume} \mathrm{is} \mathrm{represented} \mathrm{by} \mathrm{that} \mathrm{definite} \mathrm{integral}: \\ \underset{\mathrm{n}\to \infty }{\lim } \sum _{\mathrm{k}=1}^{\mathrm{n}}\mathrm{\pi }{(4-({\mathrm{x}}_{\mathrm{k}}^{*})}^2) \left(\frac{2}{\mathrm{n}}\right), \mathrm{with} {\mathrm{x}}_{\mathrm{k}}^{*}={\mathrm{x}}_{\mathrm{k}}=\mathrm{k}\left(\frac{2}{\mathrm{n}}\right) \end{gathered}$$

$$\begin{gathered} \mathrm{Suppose} \mathrm{that} \mathrm{for} \mathrm{some} \mathrm{b} > 0, \mathrm{the} \mathrm{region} \mathrm{between} \mathrm{y} = \sqrt{\mathrm{x}} \mathrm{and} \mathrm{y} = 0 \mathrm{on} [0, \mathrm{b}], \\ \mathrm{rotated} \mathrm{around} \mathrm{the} \mathrm{x}-\mathrm{axis}, \mathrm{has} \mathrm{volume} \mathrm{V} = 8\mathrm{\pi }.\mathrm{Without} \mathrm{solving} \mathrm{any} \mathrm{integrals}, \\ \mathrm{find} \mathrm{the} \mathrm{volume} \mathrm{of} \mathrm{solid} \mathrm{obtained} \mathrm{by} \mathrm{rotating} \mathrm{the} \mathrm{region} \mathrm{bounded} \mathrm{by} \mathrm{y} = \sqrt{\mathrm{x}}, \mathrm{y} = \sqrt{\mathrm{b}}, \\ \mathrm{and} \mathrm{x} = 0 \mathrm{around} \mathrm{the} \mathrm{x}-\mathrm{axis}. \end{gathered}$$

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals

$\pi {\int }_0^{\frac{\pi }{2}}{\cos }^2\left(x\right)dx and \pi {\int }_{\frac{\pi }{2}}^{\pi }{\cos }^2\left(x\right)dx$

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals 

$\pi {\int }_0^1{e}^{2x}dx and \pi {\int }_0^1(e^{2x}-1)dx$

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals  

$\pi {\int }_0^2{x}^{4}dx and \pi {\int }_0^2(x^4-16)dx$

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals 

$\pi {\int }_0^3{x}^{2}dx and \pi {\int }_0^3(9-x^2)dx$

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals 

$\pi {\int }_0^2{x}^{4}dx and \pi {\int }_0^{4}ydy$

For each pair of definite integrals in Exercises 13–18, decide which, if either, is larger, without computing any integrals.

$\mathrm{\pi }{\int }_0^4\mathrm{ydy} \mathrm{and} \mathrm{\pi }{\int }_4^8(8-\mathrm{y})\mathrm{dy}$

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.
$\mathrm{\pi }{\int }_1^3\left(16-{\left(\mathrm{x}+1\right)}^2\right)\mathrm{dx}$

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region. 

$\mathrm{\pi }{\int }_0^{\mathrm{\pi }}{\sin }^2\mathrm{xdx}$

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.  

$\mathrm{\pi }{\int }_1^3({\mathrm{x}}^2-2\mathrm{x}+1)\mathrm{dx}$

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.   

$\mathrm{\pi }{\int }_0^2\mathrm{ydy}$

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.    

$\mathrm{\pi }{\int }_1^5{\left(\frac{\mathrm{y}-1}{2}\right)}^2\mathrm{dy}$

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.   

$\mathrm{\pi }{\int }_0^4(2^2-{\left(\sqrt{\mathrm{y}}\right)}^2)\mathrm{dy}$

Write the volume of the two solids of revolution that follow in terms of definite integrals that represent accumulations of disks and/or washers. Do not compute the integrals. 

Write the volume of the two solids of revolution that follow in terms of definite integrals that represent accumulations of disks and/or washers. Do not compute the integrals.