?1. Evaluate where n is an integer and D is the region bounded by the circles with

Suppose is a continuous function defined on a rectangle (a) Write an expression for a double Riemann sum of . If , what does the sum represent? (b) Write the definition of as a limit. (c) What is the geometric interpretation of if What if takes on both positive and negative values? (d) How do you evaluate (e) What does the Midpoint Rule for double integrals say? (f) Write an expression for the average value of .

(a) How do you define \(\iint_D f(x, y) d A\) if D is a bounded region that is not a rectangle? (b) What is a type I region? How do you evaluate \(\iint_D f(x, y) d A\) if D is a type I region? (c) What is a type II region? How do you evaluate \(\iint_D f(x, y) d A\) if D is a type II region? (d) What properties do double integrals have?

How do you change from rectangular coordinates to polar coordinates in a double integral? Why would you want to make the change?

If a lamina occupies a plane region and has density function , write expressions for each of the following in terms of double integrals. (a) The mass (b) The moments about the axes (c) The center of mass (d) The moments of inertia about the axes and the origin

Let be a joint density function of a pair of continuous random variables and . (a) Write a double integral for the probability that lies between and and lies between and (b) What properties does possess? (c) What are the expected values of and

Write an expression for the area of a surface with equation

(a) Write the definition of the triple integral of over a rectangular box (b) How do you evaluate (c) How do you define if is a bounded solid region that is not a box? (d) What is a type 1 solid region? How do you evaluate if is such a region? (e) What is a type 2 solid region? How do you evaluate if is such a region? (f) What is a type 3 solid region? How do you evaluate if is such a region?

Suppose a solid object occupies the region and has density function . Write expressions for each of the following. (a) The mass (b) The moments about the coordinate planes (c) The coordinates of the center of mass (d) The moments of inertia about the axes

(a) How do you change from rectangular coordinates to cylindrical coordinates in a triple integral? (b) How do you change from rectangular coordinates to spherical coordinates in a triple integral? (c) In what situations would you change to cylindrical or spherical coordinates?

(a) If a transformation is given bywhat is the Jacobian of (b) How do you change variables in a double integral? (c) How do you change variables in a triple integral?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int_{-1}^2 \int_0^6 x^2 \sin (x-y) d x d y=\int_0^6 \int_{-1}^2 x^2 \sin (x-y) d y d x\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int_0^1\int_0^x\sqrt{x+y^2}\ dy\ dx=\int_0^x\int_0^1\sqrt{x+y^2}\ dx\ dy\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int_{1}^{2} \int_{3}^{4} x^{2} e^{y} d y d x=\int_{1}^{2} x^{2} d x \int_{3}^{4} e^{y} d y\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int_{-1}^1\int_0^1e^{x^2+y^2}\sin y\ dx\ dy=0\) Equation Transcription: Text Transcription: int_-1^1 int_0^1 e^x^2 + y^2 sin y dx dy = 0

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [0, 1], then \(\int_0^1 \int_0^1 f(x) f(y) d y d x=\left[\int_0^1 f(x) d x\right]^2\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int_{1}^{4} \int_{0}^{1}\left(x^{2}+\sqrt{y)} \sin \left(x^{2} y^{2}\right) d x\ d y<9\right.\) Equation Transcription: ? 9 Text Transcription: int_1^4 \int_0^1 (x^2 + sqrt y) sin (x^2 y^2) dx dy leqslant 9

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If D is the disk given by \(x^{2}+y^{2} \leqslant 4\), then \(\iint_{D} \sqrt{4-x^{2}-y^{2}} d A=\frac{16}{3} \pi\) Equation Transcription: ? ? Text Transcription: x^2 + y^2 leqslant 4 double integral_D sqrt 4-x^2 -y^2 dA =16/3 pi

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The integral represents the moment of inertia about the -axis of a solid with constant density .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The integral \(\int_{0}^{2 \pi} \int_{0}^{2} \int_{r}^{2} d z d r d \theta\) represents the volume enclosed by the cone \(z=\sqrt{x^{2}+y^{2}}\) and the plane z=2.

A contour map is shown for a function f on the square \(R=[0,3] \times[0,3]\). Use a Riemann sum with nine terms to estimate the value of \(\iint_{R} f(x, y) d A\). Take the sample points to be the upper right corners of the squares.

Use the Midpoint Rule to estimate the integral in Exercise 1 .

Calculate the iterated integral. \(\int_{1}^{2} \int_{0}^{2}\left(y+2 x e^{y}\right) d x d y\)

Calculate the iterated integral. \(\int_0^1 \int_0^1 y e^{x y} d x d y\)

Calculate the iterated integral. \(\int_0^1\int_0^x\cos\left(x^2\right) \ dy \ dx\)

Calculate the iterated integral. \(\int_0^1 \int_x^{e^x} 3 x y^2 d y d x\)

Calculate the iterated integral. \(\int_0^{\pi}\int_0^1\int_0^{\sqrt{1-y^2}}y\ \sin x\ dz\ dy\ dx\) Equation Transcription: Text Transcription: int_0^pi int_0^1 int_0^sqrt1- y^2 y sin x dz dy dx

Calculate the iterated integral.

Write \(\iint_R f(x, y) d A\) as an iterated integral, where R is the region shown and f is an arbitrary continuous function on R.

Write as an iterated integral, where is the region shown and is an arbitrary continuous function on .

The cylindrical coordinates of a point are . Find the rectangular and spherical coordinates of the point.

The rectangular coordinates of a point are . Find the cylindrical and spherical coordinates of the point.

The spherical coordinates of a point are . Find the rectangular and cylindrical coordinates of the point.

Identify the surfaces whose equations are given. (a) (b)

Write the equation in cylindrical coordinates and in spherical coordinates. (a) (b)

Sketch the solid consisting of all points with spherical coordinates \((p, \ \theta, \ \phi)\) such that \(0 \leqslant \theta \leqslant \pi / 2, \ 0 \leqslant \phi \leqslant \pi / 6\) and \(0 \leqslant \rho \leqslant 2 \ \cos \ \phi\).

Describe the region whose area is given by the integral \(\int_{0}^{\pi / 2} \int_{0}^{\sin 2 \theta} r d r d \theta\) Equation Transcription: Text Transcription: int_0^pi/2 int_0^sin 2theta r dr d theta

Describe the solid whose volume is given by the integral \(\int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{1}^{2} p^{2} \sin \phi d \rho d \phi d \theta\) and evaluate the integral.

Calculate the iterated integral by first reversing the order of integration. \(\int_0^1\int_x^1\cos\left(y^2\right)\ dy\ dx\) Equation Transcription: Text Transcription: int_0^1 int_x^1 cos(y^2) dy dx

Calculate the iterated integral by first reversing the order of integration. \(\int_0^1 \int_{\sqrt{y}}^1 \frac{y e^{x^2}}{x^3} d x d y\)

Calculate the value of the multiple integral. , where

Calculate the value of the multiple integral. \(\iint_{D} x y \ d A\), where \(D=\left\{(x, y) \mid 0 \leqslant y \leqslant 1, \ y^{2} \leqslant x \leqslant y+2\right\}\)

Calculate the value of the multiple integral. \(\iint_{D} \frac{y}{1+x^{2}} d A\),where D is bounded by \(y=\sqrt{x}, y=0, x=1\) Equation Transcription: ? Text Transcription: double integral_D y/1+x^2 dA y=sqrt x, y=0, x=1

Calculate the value of the multiple integral. \(\iint_{D} \frac{1}{1+x^{2}} d A\), where D is the triangular region with vertices \((0,0),(1,1)\), and \((0, 1)\) Equation Transcription: ? Text Transcription: double integral_D 1/1+x^2 dA (0 ,0), (1, 1) (0, 1)

Calculate the value of the multiple integral. \(\iint_{D} y d A\), where D is the region in the first quadrant bounded by the parabolas \(x=y^{2}\) and \(x=8-y^{2}\) Equation Transcription: ? Text Transcription: Double Integral_D ydA x = y^2 x = 8 - y^2

Calculate the value of the multiple integral. , where is the region in the first quadrant that lies above the hyperbola and the line and below the line

Calculate the value of the multiple integral. \(\iint_{D}\left(x^{2}+y^{2}\right)^{3 / 2} d A\), where D is the region in the first quadrant bounded by the lines \(y=0\) and \(y=\sqrt{3} x\) and the circle \(x^{2}+y^{2}=9\) Equation Transcription: ? Text Transcription: Double Integral_D (x^2 + y^2)^3/2 dA y = 0 y = sqrt 3x x^2 + y^2 = 9

Calculate the value of the multiple integral. \(\iint_{D} x d A\), where D is the region in the first quadrant that lies between the circles \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=2\) Equation Transcription: ? Text Transcription: Double Integral D xdA x^2 + y^2 = 1 x^2 + y^2 = 2

Calculate the value of the multiple integral. , where

Calculate the value of the multiple integral. , where is the solid tetrahedron with vertices , and

Calculate the value of the multiple integral. \(\iint_{E} y^{2} z^{2} d V\), where E is bounded by the paraboloid \(x=1-y^{2}-z^{2}\) and the plane \(x=0\) Equation Transcription: ? Text Transcription: Double Integral_E y^2 z^2 dV x = 1 - y^2 - z^2 x = 0

Calculate the value of the multiple integral. \(\iiint_{E} z \ d V\), where E is bounded by the planes \(y=0, \ z=0\), \(x+y=2\) and the cylinder \(y^{2}+z^{2}=1\) in the first octant

Calculate the value of the multiple integral. \(\iiint_{E} y z d V\), where E lies above the plane \(z=0\), below the plane \(z=y\), and inside the cylinder \(x^{2}+y^{2}=4\) Equation Transcription: ? Text Transcription: Triple Integral_E yzdV z = 0 z = y x^2 + y^2 = 4

Calculate the value of the multiple integral. \(\iiint z^{3} \sqrt{x^{2}+y^{2}+z^{2}} d V\), where H is the solid hemisphere that lies above the xy-plane and has center the origin and radius 1 Equation Transcription: ? Text Transcription: Triple Integral_z^3 sqrt x^2 + y^2 + z^2 dV

Find the volume of the given solid. Under the paraboloid z = x2 + 4y2 and above the rectangle R = [0, 2] X [1, 4]

Under the surface z = x2y and above the triangle in the xy-plane with vertices (1, 0), (2, 1), and (4, 0)

The solid tetrahedron with vertices (0, 0, 0), (0, 0, 1), (0, 2, 0), and (2, 2, 0)

Bounded by the cylinder x2 + y2 = 4 and the planes z = 0 and y + z = 3

One of the wedges cut from the cylinder x2 + 9y2 = a2 by the planes z = 0 and z = mx

Above the paraboloid z = x2 + y2 and below the half-cone z =

Consider a lamina that occupies the region D bounded by the parabola x = 1 - y2 and the coordinate axes in the first quadrant with density function p(x, y) = y. 1. Find the mass of the lamina. 2. Find the center of mass. 3. Find the moments of inertia and radii of gyration about the x- and y-axes.

A lamina occupies the part of the disk x2 + y2 ? a2 that lies in the first quadrant. 1. Find the centroid of the lamina. 2. Find the center of mass of the lamina if the density function is p(x, y) = xy2

1. Find the centroid of a solid right circular cone with height h and base radius a. (Place the cone so that its base is in the xy-plane with center the origin and its axis along the positive z-axis.) 2. If the cone has density function p(x, y, z) = find the moment of inertia of the cone about its axis (the z-axis).

Find the area of the part of the cone z2 = a2(x2 + y2) between the planes z = 1 and z = 2.

Find the area of the part of the surface z = x2 + y that lies above the triangle with vertices (0, 0), (1, 0), and (0, 2).

Use a computer algebra system to graph the surface \(z=x \sin y,-3 \leqslant x \leqslant 3,-\pi \leqslant y \leqslant \pi\), and find its surface area correct to four decimal places.

Use polar coordinates to evaluate

Use spherical coordinates to evaluate

If D is the region bounded by the curves y = 1 - x2 and y = ex , find the approximate value of the integral y 2 dA. (Use a graph to estimate the points of intersection of the curves.)

Use a computer algebra system to find the center of mass of the solid tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 2, 0), (0, 0, 3) and density function p(x, y, z) = x2 + y2 + z2

The joint density function for random variables X and Y is \(f(x, y)=\left\{\begin{array}{ll} C(x+y) & \text { if } 0 \leqslant x \leqslant 3,0 \leqslant y \leqslant 2 \\ 0 & \text { otherwise } \end{array}\right.\) (a) Find the value of the constant C. (b) Find \(P(X \leqslant 2, Y \geqslant 1\). (c) Find \(P(X+Y \leqslant 1)\).

A lamp has three bulbs, each of a type with average lifetime 800 hours. If we model the probability of failure of a bulb by an exponential density function with mean 800, find the probability that all three bulbs fail within a total of 1000 hours.

Rewrite the integral as an iterated integral in the order dx dy dz.

Give five other iterated integrals that are equal to

Use the transformation \(u=x-y, \ v=x+y\) to evaluate \(\int_{\mathrm{R}} \int \frac{x-y}{x+y} \ d A\) where R is the square with vertices (0, 2), (1, 1), (2, 2), and (1, 3).

Use the transformation \(x=u^{2}, \ y=v^{2}, \ z=w^{2}\) to find the volume of the region bounded by the surface \(\sqrt{x}+\sqrt{y}+\sqrt{z}=1\) and the coordinate planes.

Use the change of variables formula and an appropriate transformation to evaluate where R is the square with vertices (0, 0), (1, 1), (2, 0), and (1, -1).

1. Evaluate where n is an integer and D is the region bounded by the circles with center the origin and radii r and R, 0 < r < R. 2. For what values of n does the integral in part (a) have a limit as r 3. Find where E is the region bounded by the spheres with center the origin and radii r and R, 0 < r < R. 4. For what values of n does the integral in part (c) have a limit as r

If \([[x]]\) denotes the greatest integer in x, evaluate the integral \(\iint_{R} \ [[x+y]] \ d A\) where \(R=\{(x, \ y) \mid 1 \leqslant x \leqslant 3, \ 2 \leqslant y \leqslant 5\}\)

Evaluate the integral \(\int_0^1 \int_0^1 e^{\max \left\{x^2, y^2\right\}} d y d x\) where \(\max \left\{x^2, y^2\right\}\) means the larger of the numbers \(x^2\) and \(y^2\).

Find the average value of the function f(x) = cos (t2) dt on the interval [0, 1].

Show that \(\int_0^2 \int_0^x 2 e^{x^2-y^2} d y d x=\int_0^2 \int_y^{4-y} e^{x y} d x d y\)

The double integral \(\int_{0}^{1} \int_{0}^{1} \frac{1}{1-x y} \ d x \ d y\) is an improper integral and could be defined as the limit of double integrals over the rectangle \([0, \ t] \times[0, \ t] \text { as } t \rightarrow 1^{-}\). But if we expand the integrand as a geometric series, we can express the integral as the sum of an infinite series. Show that \(\int_{0}^{1} \int_{0}^{1} \frac{1}{1-x y} \ d x \ d y=\sum_{n=1}^{\infty} \frac{1}{n^{2}}\)

Leonhard Euler was able to find the exact sum of the series in Problem 5. In 1736 he proved that In this problem we ask you to prove this fact by evaluating the double integral in Problem 5. Start by making the change of variables y = y = This gives a rotation about the origin through the angle /4. You will need to sketch the corresponding region in the uv-plane. [Hint: If, in evaluating the integral, you encounter either of the expressions (1 - sin )/cos or (cos )/(1 + sin ), you might like to use the identity cos = sin((/2) - ) and the corresponding identity for sin .]

1. Show that (Nobody has ever been able to find the exact value of the sum of this series.) 2. Show that Use this equation to evaluate the triple integral correct to two decimal places.

Show that \(\int_{0}^{\infty} \frac{\arctan \pi x-\arctan x}{x} d x=\frac{\pi}{2} \ln \pi\) by first expressing the integral as an iterated integral.

1. Show that when Laplace’s equation is written in cylindrical coordinates, it becomes 2. Show that when Laplace’s equation is written in spherical coordinates, it becomes

(a) A lamina has constant density \(\rho\) and takes the shape of a disk with center the origin and radius R. Use Newton's Law of Gravitation (see Section 13.4) to show that the magnitude of the force of attraction that the lamina exerts on a body with mass m located at the point (0,0, d) on the positive z-axis is \(F=2 \pi G m \rho d\left(\frac{1}{d}-\frac{1}{\sqrt{R^2+d^2}}\right)\) [Hint: Divide the disk as in Figure 15.3.4 and first compute the vertical component of the force exerted by the polar subrectangle \(R_{i j}\). ] (b) Show that the magnitude of the force of attraction of a lamina with density \(\rho\) that occupies an entire plane on an object with mass m located at a distance d from the plane is \(F=2 \pi G m \rho\) Notice that this expression does not depend on d.

If f is continuous, show that f(t) dt dz dy = (x - t)2f(t) dt

Evaluate

The plane a > 0, b > 0, c > 0 Cuts the solid ellipsoid into two pieces. Find the volume of the smaller piece.