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calculus

Questions and Answers of Calculus

CalculateWhere R = {(x, y): 0 ( x ( 1, 0 ( y ( 1}. This integral represents the volume of a certain solid. Sketch this solid and calculate its volume from elementary principles.
CalculateWhere R = {(x, y): 0 ( x ( 2, 0 ( y ( 1}.This integral represents the volume of a certain solid. Sketch this solid and calculate its volume from elementary principles.
Let R be the rectangle shown in Figure 8. For the indicated partition into 12 equal squares, calculate the smallest and largest Riemann sums forAnd thereby obtain numbers c and C such that
Recall that [[x] is the greatest integer function. For R of(a) (b)
Let ((x, y) 1 if both x and y are rational numbers, and let ((x, y) = 0 otherwise. Show that ((x, y) is not integrable over the rectangle R in Figure 8.
Use the two graphs in Figure 9 to approximateR = {(x, y): 0 ( x ( 4, 0 ( 4, 0 ( y ( 4}
Suppose that R = {(x, y): 0 ( x ( 2, 0 ( y ( 2},R1 = {(x, y): 0 ( x ( 2, 0 ( y ( 1}, andR2 = {(x, y): 0 ( x ( 2, 1 ( y ( 2}. Suppose, in addition, thatAnd Use the properties of integrals to evaluate
In Problems 1-3, R = {(x, y): 0 ( x ( 6, 0 s y ( 4} and P is the partition of R into six equal squares by the lines x = 2, x = 4, and y = 2. Approximate((x, y) dA by calculating the corresponding
In Problems 1-3, evaluate each integral.1.2. 3.
EvaluateWhere S is the region bounded by y = sin x and y = 0 between x = 0 and x = (.
EvaluateWhere S is the region bounded by x2 + z = 1 and y2 + z = 1 and the xy-plane.
Find the center of mass of the rectangular lamina bounded by x = 1, x = 3, y = 0, and y = 2 if the density is ((x, y) = xy2.
Find the area of the surface of the cylinder z2 + y2 = 9 lying in the first octant between the planes y = x and y = 2x.
Evaluate by changing to cylindrical or spherical coordinates.(a)(b)
Find the center of mass of the homogeneous lamina bounded by the cardioids r = 4(1 + sin ().
Use a transformation to evaluate the integralWhere R is the rectangle with vertices (0, 0), ((/2, -(/2), ((, 0), and ((/2, (/2).
In Problem 1-3, rewrite the iterated integral with the indicated order of integration. Make a sketch first.1.2. 3.
Write the triple iterated integrals for the volume of a sphere of radius a in each case. (a) Cartesian coordinates (b) Cylindrical coordinates (c) Spherical coordinates
In Problems 1-3, find parametric equations for the given curve. (Be sure to give the domain for the parameter t) 1. The circle centered at the origin having radius 3 2. The circle centered at (2, 1)
Use the arc length formula to find the length of the curve in Problem 6. That part of the line y = 9 - x that is in the first quadrant with an orientation that is down and to the right
In Problems 1-3, find the gradient of the given function.1. ((x, y) = x sin x + y cos y2. ((x, y) = xe-xy + yexy3. ((x, y, z) = x2 + y2 + z2
Evaluate the integrals in Problems 1-3.1.2. 3.
The integral in Problem 22 represents the volume of some region in three-space. What is this region?
Find the surface area of that part of the paraboloid z = 144 - x2 - y2 that lies above the plane z = 36.
Find a unit normal vector to the graph of x2 + y2 + z2 = 169 at the point (3, 4, 12)?
In Problems 1-5, evaluate each of the iterated integrals.1.2. 3. 4. 5.
In Problems 1-3, evaluate the indicated double integral over R.1.2. 3.
In Problems 1-3, find the volume under the surface in each figure.1.2. 3.
In Problems 1-3, sketch the solid whose volume is the indicated iterated integral.1.2. 3.
In Problems 1-3, find the volume of the given solid. First, sketch the solid; then estimate its volume; finally, determine its exact volume.1. Solid under the plane z = x + y + 1 over R = {(x, y): 0
Show that if ((x, y) = g(x)h(y) then
Use Problem 33 to evaluateShow that if ((x, y) = g(x)h(y) then
Evaluate
Find the volume of the solid trapped between the surface z = cos x cos y and the xy-plane, where -( ( x ( (, -( ( y ( (.
In Problems 1-3, evaluate each iterated integral.1.2.3.
EvaluateReverse the order of integration.
Prove the Cauchy-Schawarz Inequality for Integrals:Consider the double integral of F(x, y) = [((x)g(y) - ((y)g(x)]2 Over the rectangle R = {(x, y): ( ( x ( b, ( ( y ( b}.
Suppose that ( is increasing on [(, b] andProve thatAnd give a physical interpretation of this result. Let F(x, y) = [y - x][((y) - ((x)] and use the hint of Problem 41.Consider the double integral
Evaluate the iterated integrals in Problems 1-3.1.2. 3.
In Problems 1-3, evaluate the given double integral by changing it to an iterated integral.1.S is the region between y = x2 and y = (x.2. If S is the region between y = x and y = 3x - x2.3. If S is
In Problems 1-3, sketch the indicated solid. Then find its volume by an iterated integration. 1. Tetrahedron bounded by the coordinate planes and the plane z = 6 - 2x - 3y 2. Tetrahedron bounded by
In Problems 1-3, write the given iterated integral as an iterated integral with the order of integration interchanged. Hint Begin by sketching a region S and representing it in two ways as in Example
EvaluateWhere S is the region shown in Figure 16.
EvaluateWhere S = {(x, y): 1 ( x2 + y2 ( 4}. Use symmetry to reduce the problem to evaluating Where S1 and S2 are as in Figure 18.
EvaluateWhere S is the annulus {(x, y): 1 ( x2 + y2 ( 4}. Done without thinking, this problem is hard; using symmetry, it is trivial.
EvaluateWhere S is the region bounded by y = (x, y = 2, and x = 0. If one order of integration does not work, try the other.
EvaluateWhere S is the region between the ellipse x2 + 2y2 = 4 and the circle x2 + y2 = 4.
Figure 19 shows a contour map for the depth of a river between a dam and a bridge. Approximate the volume of water between the dam and the bridge. Hint: Slice the river into eleven 100-feet sections
Suppose that ((x, y) is a continuous function defined on a region R that is closed and bounded. Show that there is an ordered pair ((, b) in R such thatThis result is called the Mean Value Theorem
In Problems 1-3, evaluate the iterated integrals.1.2.3.
In Problems 1-3, an iterated integral in polar coordinates is given. Sketch the region whose area is given by the iterated integral and evaluate the integral, thereby finding the area of the
In Problems 1-3, evaluate by using polar coordinates Sketch the region of integration first.1.Where S is the region enclosed by x2 + y2 = 4 2. Where S is the first quadrant sector of the circle x2 +
Find the volume of the solid in the first octant under the paraboloid z = x2 + y2 and inside the cylinder x2 + y2 = 9 by using polar coordinates.
Using polar coordinates, find the volume of the solid bounded above by 2x2 + 2y2 + z2 = 18, below by z = 0, and laterally by x2 + y2 = 4.
Switch to rectangular coordinates and then evaluate
LetWhere S is the region inside the circle x2 + y2 = 4(2.(a) Without calculation, determine the sign of V.(b) Evaluate V.(c) Evaluate W.
The centers of two spheres of radius ( are 2b units apart with b ( (. Find the volume of their intersection in terms of d = a - b.
The depth (in feet) of water distributed by a rotating lawn sprinkler in an hour is ke-2/10, 0 ( r ( 10, where r is the distance from the sprinkler and k is a constant. Determine k if 100 cubic feet
Find the volume of the solid cut from the sphere r2 + z2 ( (2 by the cylinder r = ( sin (.
Find the volume of the wedge cut from a tall right circular cylinder of radius a by a plane through a diameter of its base and making an angle ( (0 < ( < (/2) with the base.
Consider the ring A of height 2h obtained from a sphere of radius a when a hole of radius c(c
There is a simple explanation for the remarkable result in Problem 35. Show that a horizontal plane that intersects the region in Figure 12 and a sphere of radius b next to it will intersect in equal
Show that
Recall the formula A = 1/2 r2( for the area of the sector of a circle of radius r and central angle ( radians. Use this to obtain the formulaFor the area of the polar rectangle {(r, (): r1 ( r ( r2,
Show thatFor all ( and for all ( > 0. Use the result of Example 5.Show that
In Problems 1-3, find the area of the given region S by calculatingBe sure to make a sketch of the region first. 1. S is the region inside the circle r = 4 cos ( and outside the circle r = 2. 2. S is
In Problems 1-3, find the mass m and center of mass (x̅, y̅) of the lamina bounded by the given curves and with the indicated density 1. x = 0, x = 4, y = 0, y = 3; ((x, y) = y + 1 2. y = 0, y =
In Problems 1-3, find the moments of inertia Ix, Ix, and Iz for the lamina bounded by the given curves and with the indicated density (.1. y (x, x = 9, y = 0; ((x, y) x + y2. y = x2, y = 4; ((x, y) =
In Problems 1-3, an iterated integral is given either in rectangular or polar coordinates. The double integral gives the mass of some lamina R. Sketch the lamina R and determine the density (. Then
Find the moment of inertia and radius of gyration of a homogeneous (8 a constant) circular lamina of radius a with respect to a diameter.
Show that the moment of inertia of a homogeneous rectangular lamina with sides of length a and b about a perpendicular axis through its center of mass isI = 1/12((3b + (b3)Here k is the constant
Find the moment of inertia of the lamina of Problem 23 about a line tangent to its boundary. Let the circle be r = 2a sin 8; then the tangent line is the x-axis. Formula 113 at the back of the book
Consider the lamina S of constant density k bounded by the cardioid r = a(1 + sin (), as shown in Figure 7. Find its center of mass and moment of inertia with respect to the x-axis.
Find the center of mass of that part of the cardioid of Problem 26 that is outside the circle r = a.Consider the lamina S of constant density k bounded by the cardioid r = a(1 + sin (), as shown in
Parallel Axis Theorem Consider a lamina S of mass m together with parallel lines L and L( in the plane of S. the line L passing through the center of mass of S. Show that if I and I' are the moments
Refer to the lamina of Problem 13, for which we found Iy = 5(5/12. Find (a) m (b) x̅ (c) IL Where L is a line through (x̅, y̅) parallel to the y-axis.
The Parallel Axis theorem also holds for lines that are perpendicular to a lamina. Use this fact to find the moment of inertia of the rectangular lamina of Problem 24 about an axis per pendicular to
Let S1 and S2 be disjoint laminas in the xy-plane of mass m1 and m2 with centers of mass (xÌ…1, yÌ…1) and (xÌ…2, yÌ…2). Show that the center of mass (xÌ…, yÌ…) of the combined
Let S1 and S2 be the homogeneous circular laminas of radius ( and t( (t > 0) centered at (-(, () and (t(, 0), respectively. Use Problem 17 to find the center of mass of S1 ( S2.
Let S be a lamina in the xy-plane with center of mass at the origin, and let L be the line ax + by = 0, which goes through the origin. Show that the (signed) distance d of a point (x, y) from L is d
For the lamina of Example 3, find the equation of the balance line that makes an angle of 135° with the positive x-axis. Write your answer in the form Ax + By = C.Find the center of mass of a lamina
In Problems 1-3, find the area of the indicated surface. Make a sketch in each case. 1. The part of the plane 3x + 4y + 6z = 12 that is above the rectangle in the xy-plane with vertices (0, 0), (2,
Figure 8 shows the Engineering Building at southern Illinois University Edwardsville. The spiral staircase, visible in theMiddle of the photo, is in the shape of a right circular cylinder with
Problems 1-3 are related to Example 3.1. Consider that part of the sphere x2 + y2 + z2 = a2 between the planes z = h1 and z = h2, where 0 ( h1 2. Show that the polar cap (Figure 9) on a sphere of
Let S be a planar region in three-space, and let Sxy, Sxz, and Syz be the projections on the three coordinate planes (Figure 10). Show that[A(S)]2 = [A(Sxy)]2 + [A(Sxz)]2 + [A(Syz)]2
Assume that the region S of Figure 10 lies in the plane z = ((x, y) = ax + by + c and that S is above the xy-plane. Show that the volume of the solid cylinder under S is A(Sxy) ((xÌ…, yÌ…),
Joe's house has a rectangular base with a gable roof, and Alex's house has the same base with a pyramid-type roof (see Figure 11). The slopes of all parts of both roofs are the same. Whose roof has
Show that the surface area of a nonvertical plane over a region S in the xy-plane is A(S) sec y where y is the acute angle between a normal vector to the plane and the positive z-axis.
Let ( = ((x, y, ((x, y)) be the acute angle between the z-axis and a normal vector to the surface z = ((x, y) at the point (x, y, ((x, y)) on the surface. Show that sec ( = ((x2 + (y2 + 1. (This
In Problems 1-2, find the surface area of the given surface If an integral cannot be evaluated using the Second Fundamental Theorem of Calculus, then use the Parabolic Rule with n = 10.1. The
In Problems 1-3, evaluate the iterated integrals.1.2. 3.
In Problems 1-3, sketch the solid S. Then write an iterated integral for1. S = {(x, y, z): 0 ( x ( 1, 0 ( y ( 3, 0 ( z ( 1/6(12 - 3x - 2y)} 2. S = {(x, y, z): 0 ( x ( (4 - y2, 0 ( y ( 2, 0 ( z (
In Problems 1-3, use triple iterated integrals to find the indicated quantities. 1. Volume of the solid in the first octant bounded by y = 2x2 and y + 4z = 8 2. Volume of the solid in the first
In Problems 1-3, write the given iterated integral as an iterated integral with the indicated order of integration.1.2. 3.
Consider the solid (Figure 8) in the first octant cut off from the square cylinder with sides x = 0, x = 1, z = 0, and z = 1 by the plane 2x + y + 2z = 6. Find its volume in three ways. (a) Hard way:
Assuming that the density of the solid of Figure 8 is a constant k, find the moment of inertia of the solid with respect to the y-axis.
If the temperature at (x, y, z) is T(x, y, z) = 30 - z degrees, find the average temperature of the solid of Figure 8.
Find the center of mass of the homogeneous solid in Figure 8.
Consider the solid (Figure 9) in the first octant cut off from the square cylinder with sides x = 0, x = 1, y = 0, and y = 1 by the plane x + y + z = 4. Find its volume in three ways.(a) Hard way: by
Find the center of mass of the homogeneous solid in Figure 9.

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