3.6.E: Definite Integrals (Exercises)
- Page ID
- 78226
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
Evaluate each of the following iterated integrals.
(a) \(\int_{1}^{3} \int_{0}^{2} 3 x y^{2} d y d x\)
(b) \(\int_{0}^{\frac{\pi}{2}} \int_{0}^{\pi} 4 x \sin (x+y) d y d x\)
(c) \(\int_{-2}^{2} \int_{-1}^{1}\left(4-x^{2} y^{2}\right) d x d y\)
(d) \(\int_{0}^{2} \int_{0}^{1} e^{x+y} d x d y\)
- Answer
-
(a) \(\int_{1}^{3} \int_{0}^{2} 3 x y^{2} d y d x=32\)
(c) \(\int_{-2}^{2} \int_{-1}^{1}\left(4-x^{2} y^{2}\right) d x d y=\frac{256}{9}\)
Exercise \(\PageIndex{2}\)
Evaluate the following definite integrals over the given rectangles.
(a) \(\iint_{D}\left(y^{2}-2 x y\right) d x d y, D=[0,2] \times[0,1]\)
(b) \(\iint_{D} \frac{1}{(x+y)^{2}} d x d y, D=[1,2] \times[1,3]\)
(c) \(\iint_{D} y e^{-x} d x d y, D=[0,1] \times[0,2]\)
(d) \(\iint_{D} \frac{1}{2 x+y} d x d y, D=[1,2] \times[0,1]\)
- Answer
-
(a) \(\iint_{D}\left(y^{2}-2 x y\right) d x d y=-\frac{4}{3}\)
(c) \(\iint_{D} y e^{-x} d x d y=2\left(1-e^{-1}\right.\)
Exercise \(\PageIndex{3}\)
For each of the following, evaluate the iterated integrals and sketch the region of integration.
(a) \(\int_{0}^{2} \int_{0}^{y}\left(x y^{2}-x^{2}\right) d x d y\)
(b) \(\int_{0}^{1} \int_{x^{4}}^{x^{2}}\left(x^{2}+y^{2}\right) d y d x\)
(c) \(\int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}}\left(4-x^{2}-y^{2}\right) d y d x\)
(d) \(\int_{0}^{1} \int_{0}^{y^{2}} x y e^{-x-y} d x d y\)
- Answer
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(a) \(\int_{0}^{2} \int_{0}^{y}\left(x y^{2}-x^{2}\right) d x d y=\frac{16}{15}\)
(c) \(\int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}}\left(4-x^{2}-y^{2}\right) d y d x=2 \pi\)
Exercise \(\PageIndex{4}\)
Find the volume of the region beneath the graph of \(f(x, y)=2+x^{2}+y^{2}\) and above the rectangle \(D=[-1,1] \times[-2,2] \).
Exercise \(\PageIndex{5}\)
Find the volume of the region beneath the graph of \(f(x, y)=4-x^{2}+y^{2}\) and above the region \(D=\{(x, y): 0 \leq x \leq 2,-x \leq y \leq x\}\). Sketch the region \(D\).
- Answer
-
\(\frac{32}{3}\)
Exercise \(\PageIndex{6}\)
Evaluate \(\iint_{D} x y d x d y\), where \(D\) is the region bounded by the \(x\)-axis, the \(y\)-axis, and \(D\) the line \(y=2-x\).
Exercise \(\PageIndex{7}\)
Evaluate \(\iint_{D} e^{-x^{2}} d x d y\) where \(D=\{(x, y): 0 \leq y \leq 1, y \leq x \leq 1\}\).
- Answer
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\(\iint_{D} e^{-x^{2}} d x d y=\frac{1}{2}\left(1-e^{-1}\right)\)
Exercise \(\PageIndex{8}\)
Find the volume of the region in \(\mathbb{R}^3\) described by \(x \geq 0\), \(y \geq 0\), and \(0 \leq z \leq 4-2 y-4 x\).
Exercise \(\PageIndex{9}\)
Find the volume of the region in \(\mathbb{R}^3\) lying above the \(xy\)-plane and below the surface with equation \(z=16-x^{2}-y^{2}\).
- Answer
-
\(56 \pi \)
Exercise \(\PageIndex{10}\)
Find the volume of the region in \(\mathbb{R}^3\) lying above the \(xy\)-plane and below the surface with equation \(z=4-2 x^{2}-y^{2} .\)
Exercise \(\PageIndex{11}\)
Evaluate each of the following iterated integrals.
(a) \(\int_{1}^{2} \int_{0}^{3} \int_{-2}^{2}\left(4-x^{2}-z^{2}\right) d y d x d z\)
(b) \(\int_{-2}^{3} \int_{-1}^{2} \int_{0}^{2} 3 x y z d x d y d z\)
(c) \(\int_{0}^{4} \int_{0}^{x} \int_{0}^{x+y}\left(x^{2}-y z\right) d z d y d x\)
(d) \(\int_{0}^{1} \int_{0}^{x} \int_{0}^{x+y} \int_{0}^{x+y+z} w d w d z d y d x\)
- Answer
-
(a) \(\int_{1}^{2} \int_{0}^{3} \int_{-2}^{2}\left(4-x^{2}-z^{2}\right) d y d x d z=-16\)
(c) \(\int_{0}^{4} \int_{0}^{x} \int_{0}^{x+y}\left(x^{2}-y z\right) d z d y d x=\frac{2432}{15}\)
Exercise \(\PageIndex{12}\)
Find the volume of the region in \(\mathbb{R}^3\) bounded by the paraboloids with equations \(z=3-x^{2}-y^{2}\) and \(z=x^{2}+y^{2}-5\).
- Answer
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\(16 \pi \)
Exercise \(\PageIndex{13}\)
Evaluate \(\iiint_{D} x y d x d y d z\), where \(D\) is the region bounded by the \(xy\)-plane, the \(yz\)-plane, the \(xz\)-plane, and the plane with equation \(z = 4 − x − y\).
Exercise \(\PageIndex{14}\)
If \(f(x,y,z)\) represents the density of mass at the point \((x,y,z)\) of an object occupying a region \(D\) in \(\mathbb{R}^3\), then
\[ \iiint_{D} f(x, y, z) d x d y d z \nonumber \]
is the total mass of the object and the point \((\bar{x}, \bar{y}, \bar{z})\), where
\[ \bar{x}=\frac{1}{m} \iiint_{D} x f(x, y, z) d x d y d z , \nonumber \]
\[ \bar{y}=\frac{1}{m} \iiint_{D} y f(x, y, z) d x d y d z , \nonumber \]
and
\[ \bar{z}=\frac{1}{m} \iiint_{D} z f(x, y, z) d x d y d z , \nonumber \]
is called the center of mass of the object. Suppose \(D\) is the region bounded by the planes \(x=0, y=0, z=0\), and \(z=4-x-2y\).
(a) Find the total mass and center of mass for an object occupying the region \(D\) with mass density given by \(f(x, y, z)=1\).
(b) Find the total mass and center of mass for an object occupying the region \(D\) with mass density given by \(f(x, y, z)=z\).
- Answer
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(a) Mass: \(\frac{16}{3}\); center of mass: \(\left(1, \frac{1}{2}, 1\right)\)
(b) Mass: \(\frac{16}{3}\); center of mass: \(\left(\frac{4}{5}, \frac{2}{5}, \frac{8}{5}\right)\)
Exercise \(\PageIndex{15}\)
If \(X\) and \(Y\) are points chosen at random from the interval [0,1], then the probability that \((X,Y)\) lies in a subset \(D\) of the unit square \([0,1] \times[0,1]\) is \(\iint_{D} d x d y\).
(a) Find the probability that \(X \leq Y\).
(b) Find the probability that \(X+Y \leq 1\).
(c) Find the probability that \(XY \geq \frac{1}{2} \).
Exercise \(\PageIndex{16}\)
If \(X\), \(Y\), and \(Z\) are points chosen at random from the interval [0,1], then the probability that \((X,Y,Z)\) lies in a subset \(D\) of the unit cube \([0,1] \times[0,1] \times[0,1]\) is \(\iiint_{D} d x d y d z\).
(a) Find the probability that \(X \leq Y \leq Z \).
(b) Find the probability that \(X+Y+Z \leq 1\).
- Answer
-
(a) \(\frac{1}{6}\)
(b) \(\frac{1}{6}\)