7.E: Integration (Exercises)
- Last updated
- Nov 10, 2020
- Save as PDF
\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}These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.
7.1: Two Examples
Ex 7.1.1Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=2t+2$, and that at $t=1$ the object is at position 5. Find the position of the object at $t=2$. (answer)
Ex 7.1.2Suppose an object moves in a straight line so that its speed at time $t$ is given by $\ds v(t)=t^2+2$, and that at $t=0$ the object is at position 5. Find the position of the object at $t=2$. (answer)
Ex 7.1.3By a method similar to that in example 7.1.2, find the area under $y=2x$ between $x=0$ and any positive value for $x$. (answer)
Ex 7.1.4By a method similar to that in example 7.1.2, find the area under $y=4x$ between $x=0$ and any positive value for $x$. (answer)
Ex 7.1.5By a method similar to that in example 7.1.2, find the area under $y=4x$ between $x=2$ and any positive value for $x$ bigger than 2. (answer)
Ex 7.1.6By a method similar to that in example 7.1.2, find the area under $y=4x$ between any two positive values for $x$, say $a < b$. (answer)
Ex 7.1.7Let $\ds f(x)=x^2+3x+2$. Approximate the area under the curve between $x=0$ and $x=2$ using 4 rectangles and also using 8 rectangles. (answer)
Ex 7.1.8Let $\ds f(x)=x^2-2x+3$. Approximate the area under the curve between $x=1$ and $x=3$ using 4 rectangles. (answer)
7.2: The Fundamental Theorem of Calculus
Find the antiderivatives of the functions:
Ex 7.2.1 8\sqrt{x} (answer)
Ex 7.2.2 3t^2+1 (answer)
Ex 7.2.3 4/\sqrt{x} (answer)
Ex 7.2.4 2/z^2 (answer)
Ex 7.2.5 7s^{-1} (answer)
Ex 7.2.6 (5x+1)^2 (answer)
Ex 7.2.7 (x-6)^2 (answer)
Ex 7.2.8 x^{3/2} (answer)
Ex 7.2.9 {2\over x\sqrt x} (answer)
Ex 7.2.10 |2t-4| (answer)
Compute the values of the integrals:
Ex 7.2.11 \int_1^4 t^2+3t\,dt (answer)
Ex 7.2.12 \int_0^\pi \sin t\,dt (answer)
Ex 7.2.13 \int_1^{10} {1\over x}\,dx (answer)
Ex 7.2.14 \int_0^5 e^x\,dx (answer)
Ex 7.2.15 \int_0^3 x^3\,dx (answer)
Ex 7.2.16 \int_1^2 x^5\,dx (answer)
Ex 7.2.17Find the derivative of G(x)=\int_1^x t^2-3t\,dt (answer)
Ex 7.2.18Find the derivative of G(x)=\int_1^{x^2} t^2-3t\,dt (answer)
Ex 7.2.19Find the derivative of G(x)=\int_1^x e^{t^2}\,dt (answer)
Ex 7.2.20Find the derivative of G(x)=\int_1^{x^2} e^{t^2}\,dt (answer)
Ex 7.2.21Find the derivative of G(x)=\int_1^x \tan(t^2)\,dt (answer)
Ex 7.2.22Find the derivative of G(x)=\int_1^{x^2} \tan(t^2)\,dt (answer)
7.3: Some Properties of Integrals
Ex 7.3.1An object moves so that its velocity at time $t$ is $v(t)=-9.8t+20$ m/s. Describe the motion of the object between $t=0$ and $t=5$, find the total distance traveled by the object during that time, and find the net distance traveled. (answer)
Ex 7.3.2An object moves so that its velocity at time $t$ is $v(t)=\sin t$. Set up and evaluate a single definite integral to compute the net distance traveled between $t=0$ and $t=2\pi$. (answer)
Ex 7.3.3An object moves so that its velocity at time $t$ is $v(t)=1+2\sin t$ m/s. Find the net distance traveled by the object between $t=0$ and $t=2\pi$, and find the total distance traveled during the same period. (answer)
Ex 7.3.4Consider the function $f(x)=(x+2)(x+1)(x-1)(x-2)$ on $[-2,2]$. Find the total area between the curve and the $x$-axis (measuring all area as positive). (answer)
Ex 7.3.5Consider the function $\ds f(x)=x^2-3x+2$ on $[0,4]$. Find the total area between the curve and the $x$-axis (measuring all area as positive). (answer)
Ex 7.3.6Evaluate the three integrals: A=\int_0^3 (-x^2+9)\,dx\qquad B=\int_0^{4} (-x^2+9)\,dx\qquad C=\int_{4}^3 (-x^2+9)\,dx, and verify that $A=B+C$. (answer)
Contributors
