4.5E: Exercises
- Page ID
- 13748
This page is a draft and is under active development.
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
1. How are the definite and indefinite integrals related?
2. What constants of integration is most commonly used when evaluating definite integrals?
3. T/F: If \(f\) is a continuous function, then \(F(x) =\int_a^x f(t)\,dt\) is also a continuous function.
4. The definite integral can be used to find "the area under a curve." Give two other uses for definite integrals.
- Answer
-
Under Construction
Exercise \(\PageIndex{2}\)
Evaluate the definite integrals
1. \(\int_1^3 (3x^2-2x+1)\,dx\)
2. \(\int_0^4 (x-1)^2\,dx\)
3. \(\int_{-1}^1 (x^3-x^5)\,dx\)
4. \(\int_{\pi/2}^{\pi}\cos x\,dx\)
5. \(\int_0^{\pi/4}\sec^2 x\,dx\)
6. \(\int_1^e \frac{1}{x}\,dx\)
7. \(\int_{-1}^1 5^x \,dx\)
8. \(\int_{-2}^{-1}(4-2x^3)\,dx\)
9. \(\int_0^{\pi}(2\cos x -2\sin x)\,dx\)
10. \(\int_1^3 e^x\,dx\)
11. \(\int_0^4 \sqrt{t}\,dt\)
12. \(\int_9^{25} \frac{1}{\sqrt{t}}\,dt\)
13. \(\int_1^8 \sqrt[3]{x}\,dx\)
14. \(\int_1^2 \frac{1}{x}\,dx\)
15. \(\int_1^2 \frac{1}{x^2}\,dx\)
16. \(\int_1^2 \frac{1}{x^3}\,dx\)
17. \(\int_0^1 x\,dx\)
18. \(\int_0^1 x^2\,dx\)
19. \(\int_0^1 x^3\,dx\)
20. \(\int_0^1 x^{100}\,dx\)
21. \(\int_{-4}^4 dx\)
22. \(\int_{-10}^{-5} 3\,dx\)
23. \(\int_{-2}^2 0\,dx\)
24. \(\int_{\pi/6}^{\pi/3}\csc x \cot x\,dx\)
- Answer
-
Under Construction
Exercise \(\PageIndex{3}\)
29. Explain why:
(a) \(\int_{-1}^1 x^n\,dx=0\), when n is a positive, odd integer, and
(b) \(\int_{-1}^1x^n\,dx =2\int_0^1 x^n \,dx\) when n is a positive, even integer.
- Answer
-
Under Construction
Exercise \(\PageIndex{4}\)
Find a value c guaranteed by the Mean Value Theorem.
1. \(\int_0^2 x^2\,dx\)
2. \(\int_{-2}^2 x^2\,dx\)
3. \(\int_0^1 e^x\,dx\)
4. \(\int_0^16 \sqrt{x}\,dx\)
- Answer
-
Under Construction
Exercise \(\PageIndex{5}\)
Find the average value of the function on the given interval.
1. \(f(x) =\sin x \text{ on }[0,\pi/2]\)
2. \(y =\sin x \text{ on }[0,\pi]\)
3. \(y = x \text{ on }[0,4]\)
4. \(y =x^2 \text{ on }[0,4]\)
5. \(y =x^3 \text{ on }[0,4]\)
6. \(g(t) =1/t \text{ on }[1,e]\)
- Answer
-
Under Construction
Exercise \(\PageIndex{6}\)
A velocity function of an object moving along a straight line is given. Find the displacement of the object over the given time interval.
1. \(v(t) =-32t+20\)ft/s on [0,5].
2. \(v(t) =-32t+200\)ft/s on [0,10].
3. \(v(t) =2^t\)mph on [-1,1].
4. \(v(t) =\cos t\)ft/s on \([0,3\pi /2]\).
5. \(v(t) =\sqrt[4]{t}\)ft/s on [0,16].
- Answer
-
Under Construction
Exercise \(\PageIndex{7}\)
An acceleration function of an object moving along a straight line is given. Find the change of the object's velocity over the given time interval.
1. \(a(t) =-32\)ft/s on [0,2].
2. \(a(t) =10\)ft/s on [0,5].
3. \(a(t) =t\)ft/s\(^2\) on [0,2].
4. \(a(t) =\cos t\)ft/s\(^2\) on \([0,\pi]\).
- Answer
-
Under Construction
Exercise \(\PageIndex{8}\)
Sketch the given functions and find the area of the enclosed region.
1. \(y =2x,\, y=5x,\text{ and }x=3\).
2. \(y=-x+1,\,y=3x+6,\,x=2\text{ and }x=-1\).
3. \(y=x^2-2x+5,\,y=5x-5\).
4. \(y = 2x^2+2x-5,\,y=x^2+3x+7\).
- Answer
-
Under Construction
Exercise \(\PageIndex{9}\)
Find \(F'(x)\).
1. \(F(x) =\int_2^{x^3+x}\frac{1}{t}\,dt\)
2. \(F(x) = \int_{x^3}^0 t^3\,dt\)
3. \(F(x)=\frac{x}{x^2}(t+2)\,dt\)
4. \(F(x) =\int_{\ln x}^{e^x}\sin t\,dt\)
- Answer
-
Under Construction