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Find the volume of the indicated region. -  the region that lies under the plane x2+y4+z9=1 and above the square 0≤x,y≤43\text { the region that lies under the plane } \frac { x } { 2 } + \frac { y } { 4 } + \frac { z } { 9 } = 1 \text { and above the square } 0 \leq x , y \leq \frac { 4 } { 3 }


A) 72
B) 36
C) 8
D) 18

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Find the area of the region specified by the integral(s) . - ∫03∫x(x−3) x(x−3) (x−6) dydx\int _ { 0 } ^ { 3 } \int _ { x ( x - 3 ) } ^ { x ( x - 3 ) ( x - 6 ) } d y d x


A) 994\frac { 99 } { 4 }
B) 2614\frac { 261 } { 4 }
C) 3694\frac { 369 } { 4 }
D) 1894\frac { 189 } { 4 }

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Solve the problem. -Let D\mathrm { D } be the region bounded below by the cone z=x2+y2z = \sqrt { x ^ { 2 } + y ^ { 2 } } and above by the sphere z=81−x2−y2z = \sqrt { 81 - x ^ { 2 } - y ^ { 2 } } . Set up the triple integral in cylindrical coordinates that gives the volume of using the order of integration dzdrdθ\mathrm { dz } \mathrm { dr } \mathrm { d } \theta .


A) ∫02π∫09∫081−r2rdzdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 9 } \int _ { 0 } ^ { \sqrt { 81 - r ^ { 2 } } } r d z d r d \theta
B) ∫02π∫09/2∫081−r2rdzdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 9 / \sqrt { 2 } } \int _ { 0 } ^ { \sqrt { 81 - r ^ { 2 } } } r d z d r d \theta
C) ∫0π/2∫09/2∫081−r2rdzdrdθ\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 9 / \sqrt { 2 } } \int _ { 0 } ^ { \sqrt { 81 - r ^ { 2 } } } r d z d r d \theta
D) ∫0π/2∫09∫081−r2rdzdrdθ\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 9 } \int _ { 0 } ^ { \sqrt { 81 - r ^ { 2 } } } r d z d r d \theta

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Evaluate the cylindrical coordinate integral. - ∫3π5π∫69∫10/r11/rdzrrdrdθ\int _ { 3 \pi } ^ { 5 \pi } \int _ { 6 } ^ { 9 } \int _ { 10 / \mathrm { r } } ^ { 11 / \mathrm { r } } \mathrm { dzrrdr } \mathrm { d } \theta


A) 6π6 \pi
B) 60π60 \pi
C) 120π120 \pi
D) 12π12 \pi

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Solve the problem. -Find the moment of inertia about the yy -axis of the thin semicircular region of constant density δ=3\delta = 3 bounded by the xx -axis and the curve y=49−x2y = \sqrt { 49 - x ^ { 2 } } .


A) 2401π2401 \pi
B) 72038π\frac { 7203 } { 8 } \pi
C) 72034π\frac { 7203 } { 4 } \pi
D) 24012π\frac { 2401 } { 2 } \pi

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Solve the problem. -Let D\mathrm { D } be the region that is bounded below by the cone φ=π4\varphi = \frac { \pi } { 4 } and above by the sphere ϱ=6\varrho = 6 . Set up the triple integral for the volume of D\mathrm { D } in cylindrical coordinates.


A) ∫02π∫06∫r36−r2rdzdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 6 } \int _ { r } ^ { \sqrt { 36 - r ^ { 2 } } } r d z d r d \theta
B) ∫02π∫06/2∫036−r2rdzdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 6 / \sqrt { 2 } } \int _ { 0 } ^ { \sqrt { 36 - r ^ { 2 } } } r d z d r d \theta
C) ∫02π∫06∫036−r2rdzdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 6 } \int _ { 0 } ^ { \sqrt { 36 - r ^ { 2 } } } r d z d r d \theta
D) ∫02π∫06/2∫r36−r2rdzdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 6 / \sqrt { 2 } } \int _ { r } ^ { \sqrt { 36 - r ^ { 2 } } } r d z d r d \theta

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Solve the problem. -Find the moment of inertia about the zz -axis of a region of constant density δ\delta enclosed by the paraboloid z=16−x2−y2z = 16 - x ^ { 2 } - y ^ { 2 } and the xyx y -plane.


A) 83δπ\frac { 8 } { 3 } \delta \pi
B) 20483δπ\frac { 2048 } { 3 } \delta \pi
C) 1283δπ\frac { 128 } { 3 } \delta \pi
D) 323δπ\frac { 32 } { 3 } \delta \pi

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 Set up the iterated integral for evaluating ∬D∫f(r,θ,z) dzrdrdθ over the given region D. \text { Set up the iterated integral for evaluating } \iint _ { D } \int f ( r , \theta , z ) d z r d r d \theta \text { over the given region D. } - D\mathrm { D } is the solid right cylinder whose base is the region between the circles r=3sin⁡θr = 3 \sin \theta and r=4sin⁡θr = 4 \sin \theta , and whose top lies in the plane z=5−x−yz = 5 - x - y .


A) ∫02π∫3sin⁡θ4sin⁡θ∫05−r(cos⁡θ+sin⁡θ) f(r,θ,z) dzrdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 3 \sin \theta } ^ { 4 \sin \theta } \int _ { 0 } ^ { 5 - r ( \cos \theta + \sin \theta ) } f ( r , \theta , z ) d z r d r d \theta
B) ∫0π∫3sin⁡θ4sin⁡θ∫05−r(cos⁡θ−sin⁡θ) f(r,θ,z) dzrdrdθ\int _ { 0 } ^ { \pi } \int _ { 3 \sin \theta } ^ { 4 \sin \theta } \int _ { 0 } ^ { 5 - r ( \cos \theta - \sin \theta ) } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta
C) ∫02π∫3sin⁡θ4sin⁡θ∫05−r(cos⁡θ−sin⁡θ) f(r,θ,z) dzrdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 3 \sin \theta } ^ { 4 \sin \theta } \int _ { 0 } ^ { 5 - r ( \cos \theta - \sin \theta ) } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta
D) ∫0π∫3sin⁡θ4sin⁡θ∫05−r(cos⁡θ+sin⁡θ) f(r,θ,z) dzrdrdθ\int _ { 0 } ^ { \pi } \int _ { 3 \sin \theta } ^ { 4 \sin \theta } \int _ { 0 } ^ { 5 - r ( \cos \theta + \sin \theta ) } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta

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Find the area of the region specified in polar coordinates. -the region enclosed by the curve r=2+cos⁡θr = 2 + \cos \theta


A) 5π5 \pi
B) 92π\frac { 9 } { 2 } \pi
C) 52π\frac { 5 } { 2 } \pi
D) 3π3 \pi

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Solve the problem. -Find the mass of the region of density δ(x,y,z) =181−x2−y2\delta ( x , y , z ) = \frac { 1 } { 81 - x ^ { 2 } - y ^ { 2 } } bounded by the paraboloid z=81−x2−y2z = 81 - x ^ { 2 } - y ^ { 2 } and the xyx y -plane.


A) 6561π6561 \pi
B) 729π729 \pi
C) 9π9 \pi
D) 81π81 \pi

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Find the average value of F(x, y, z) over the given region. - F(x,y,z) =x2y6z9F ( x , y , z ) = x ^ { 2 } y ^ { 6 } z ^ { 9 } over the cube in the first octant bounded by the coordinate planes and the planes x=1x = 1 , y=1,z=1y = 1 , z = 1


A) 1192\frac { 1 } { 192 }
B) 184\frac { 1 } { 84 }
C) 1108\frac { 1 } { 108 }
D) 1210\frac { 1 } { 210 }

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Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. - z=x2+y2;0≤x≤1,0≤y≤1z = x ^ { 2 } + y ^ { 2 } ; 0 \leq x \leq 1,0 \leq y \leq 1


A) 23\frac { 2 } { 3 }
B) 13\frac { 1 } { 3 }
C) 83\frac { 8 } { 3 }
D) 43\frac { 4 } { 3 }

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Provide an appropriate response. -True or false? Consider the double integral ∫02∫36(x2+y)dxdy\int _ { 0 } ^ { 2 } \int _ { 3 } ^ { 6 } \left( x ^ { 2 } + y \right) d x d y The first step in calculating this integral involves integrating with respect to xx .

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Solve the problem. -Evaluate ∫0∞e−2x−e−8xxdx\int _ { 0 } ^ { \infty } \frac { e ^ { - 2 x } - e ^ { - 8 x } } { x } d x by writing the integrand as an integral.


A) ln⁡14\ln \frac { 1 } { 4 }
B) ln⁡2ln⁡8\frac { \ln 2 } { \ln 8 }
C) ln⁡8ln⁡2\frac { \ln 8 } { \ln 2 }
D) ln⁡4\ln 4

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Solve the problem. -Find the mass of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve y=e−8xy = e ^ { - 8 x } if δ(x,y) =xy\delta ( x , y ) = x y .


A) 1512\frac { 1 } { 512 }
B) 1192\frac { 1 } { 192 }
C) 1256\frac { 1 } { 256 }
D) 196\frac { 1 } { 96 }

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