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3.6.E: Definite Integrals (Exercises)

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    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

    (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

    \(\iint_{D} e^{-x^{2}} d x d y=\frac{1}{2}\left(1-e^{-1}\right)\)

    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

    \(16 \pi \)

    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

    (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) \(\frac{1}{6}\)

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