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# Integrals

### Surface Area of a Revolving Curve

Using the formula for the surface area of a revolving curve about the y-axis: S=∫2πx√(1 + (dx/dy)²)dy throughout a,b Find the area of the surface generated by revolving the curve about the y axis within the given boundaries x=√(2y-1) 5/8&#8804;y&#8804;1 the revolving base passes thru the point (1/2

### Surface Area of a Revolved Curve

Utilise the following formula that gives the surface area of a curve that revolves around the y-axis: S=∫2πx √ (1 + (dx/dy)²)dy throughout c, d Now calculate the area of the surface that would come about by rotating the curve around the y axis with the boundaries below: x = (1/3)y³'² - y ¹'² 1

### Surface area of revolving curve

Using the formula for the surface area of a revolving curve about the x-axis: S=&#8747;2&#960;y&#8730;(1 + (dy/dx)²)dx throughout a,b Find the area of the surface generated by revolving the curve about the x axis within the given boundaries y=&#8730;(x + 1) 1&#8804;x&#8804;5 Please be detailed, showing the compl

### Length of a curve

Using the formula for the length of a curve y=f(x) from a to b L=&#8747;&#8730;(1 + (dy/dx)²)dx Find the length of the curve: x=(y³&#8242;²&#8725;3)- y¹&#8242;² from y =1 to y=9 Hint: 1 + (dx/dy)² is a perfect square.

### Integrating to find length of a curve

Using the formula for the length of a curve y=f(x) from a to b L=∫√(1 + (dy/dx)²)dx Find the length of the curve: y=x³′² from x=0 to x=4

### Shell method of finding volume of revolving solid

The following curve and line define the boundaries of a solid generated by revolving it around the x axis. Using the shell method, find the volume of the solid y=&#8730;x y=0 y=2-x

### Shell method of finding volume of revolving solid

Using the shell method, find the volume of the solid generated by revolving the region bounded by the curve and line below about the x-axis. x=2y-y² x=y

### Integration in Finding the Volume of Rotated Solid

Determine the limits of integration and then Find the volume of the solid generated by revolving the region bounded by the line and curve about the x-axis: y=4-x² y=2-x Using the formula V=&#8747;&#960;[R(x)]²dx

### Area between two curves

The area I am looking for is the region bounded by the two functions y=x² and y=2-x between the limits (2,0) and (0,0) and bounded by the x axis and the point y=1 What is the area between these two curves? Using the formula A=&#8747;f(x)-g(x)dx

### Riemann sum area between two curves

Using the Riemann sum formula: A = &#8747; [f(y) - g(y)]dy from a to b Find the area between the two curves x=12y²-12y³ and x=2y²-2y. The y limits are 1 and zero. Please show a detailed solution. Thank you.

### How do you find the area between curves

Using the Riemann sum formula: A = ∫ [f(x) - g(x)]dx from a to b Find the area between y=1/2sec²t and y= -4sin²t between the points &#960;&#8725;3 and - &#960;&#8725;3 Please show a detailed solution. Thank you.

### Riemann Sum Area Between Two Curves

Using the Riemann sum formula: A = &#8747; [f(x) - g(x)]dx from a to b Find the area between y=e^x/2 and y=e^x Between the x values of 2ln2 and zero.

### Multiple Integration in Cartesian Coordinates

Please see the attached file for the full problem description. --- 1. Transform the given integral in Cartesian coordinates to one in polar coordinates and evaluate the polar integral. : refer to integral 5. 2. Determine the values of the given integrals, where W is the region bounded by the two spheres x^2 + y^2 + z^2 = a^

### Polar Coordinates and Change Order of Integration; Volume

Assuming r, θ are the polar coordinates, change the order of integration: ∫-pi/2-->pi/2 ∫0-->a cos θ f(r, θ ) dr dθ Find the volume of the ellipsoid: x^2/a^2 + y^2/b^2 + z^2/c^2 &#8804; 1 Let a and b be any numbers such that a^2 + b^2 =1 and f(x,y) be a continuous function of one variable. Perform the change of

### How Corporations Issue Securities : Interest Rate, Issue Cost and Company Expense, Private Placements and Public Issues

You need to choose between making a public offering and arranging a private placement. In each case the issue involves \$10 million face value of 10-year debt. You have the following data for each: A public issue: The interest rate on the debt would be 8.5 percent, and the debt would be issued at face value. The underwriti

### Evaluate the Definite Integrals

∫(pi/2 to 0) sin^4(x) dx -- do not use reduction forumulas use 1-cos2u/2=sin^2x ∫3x^3/sqrt(8-x^2) dx Integrate, integration

### Evaluate the Integrals (4 Problems)

Evaluate each of the following integrals: 1. ∫0-->2 6/(5x+2) dx 2. ∫1-->3 e^(-0.4t) dt 3..... 4.... Please see the attached file for the fully formatted problems. Integrate, Integration

### Reversing Order of Integration and Sums of Iterated Integrals

Problem 1 and 2: Sketch the region of integration, reverse the order of integration, and evaluate both iterated integrals. ∫0-->2 ∫0-->4-y2 x dx dy ∫0-->pi/2 &#8747;0-->cosx sin x dy dx Problem 3: When you reverse the order of integration, you should obtain a sum of iterated integrals. Make the reversals and

### Integration of Region Bounded by Plane

F(x,y,z)=y ; W is the region bounded by the plane x+y+z=2, the cylinder x^2 + z^2 = 1, and y=0. Integrate the given function over the indicated region W.

### Changing the Order of Integration and Finding the Volume

1. Using the integral &#8747;-1-->1 &#8747;x^2-->1 &#8747;0-->1-y dz dy dx a) Sketch the region of integration. Write the integral as an equivalent iterated integral in the order: b) dy dz dx c) dx dz dy d) dz dx dy 2. Find the volume of a wedge cut from the cylinder x^2 +y^2 =1 by planes z=-y and z=0. Please show me t

### Integral of a Function

I have a function (see attached). I need to integrate it over m from - infinity to infinity, h from - infinity to infinity. I need to apply a technique such that the integral takes a simple form, easy for integration. The main problem here as you can see is product of terms in the denominator. See the attached file.

### Inverse Trigonometric Functions, and Derivatives

7.5 Inverse trigonometric functions Find the exact value of the expression. 1) sin^-1 (SQRT3 / 2) 2) arctan(-1) 3) tan^-1 (SQRT 3) 4) cos^-1 (-1) 5) csc^-1 (2) 6) arcsin(-1/ (SQRT 2) 7) sec^-1 (SQRT 2) 8) arccos(cos 2pi) 9) tan^-1 (tan 3pi/4) 10) cos(arcsin ½) 11) sin(2 tan^-1 SQRT 2) 12) cos(tan^-1 (2) + tan

### Double Integrals : Areas Bounded by Triangles and Parabolas (5 Problems)

Please see the attached file for the fully formatted problems.

### Trigonometric Integrals and Integrate by Substitution

Evaluate the integral 1) ∫ (sin^3 (x)) (cos^2 (x)) dx 2) ∫ ( sin^4 (x)) (cos^5 (x)) dx 3) ∫ ( sin^6 (x)) (cos^3 (x)) dx 4) ∫ ( sin^3 (mx)) dx 5) ∫ (from 0 to pi/2 on top) (cos^2 (theta)) dtheta 6) ∫ (from 0 to pi/2 on top) (sin^2 (2theta)) dtheta 7) ∫ (from 0 to pi on top) (sin^4 (3t)) dt 8) ∫ (from 0

### Integrate by Substitution and Integrate by Parts (30 Problems with High Quality Solutions)

Evaluate the integral using integration by parts with the indicated choices of u and du. 1) &#8747; x ln x dx, u=ln x, du=xdx 2) &#8747; theta sec^2(theta) dtheta, u=theta, du=sec^2(theta) dtheta Evaluate the integral 1) &#8747; x cos 5x dx 2) &#8747; (x)(e)^(-x) dx 3) &#8747; re^(r/2) dr 4) &#8747; t sin 2t dt 5

### Sketch the surface of a spheroid inside a parabaloid and evaluate the surface integral.

The surface S is that part of the spheroid .... which lies inside the paraboloid az = x2 + y2; here a is a constant. Sketch the surface S and the paraboloid by drawing their intersections with the plane y = 0. Show that Z Z ..... where R is a region of the (x; y) plane you should find, and hence evaluate the surface integra

### Integration by Substitution Results

Please and explain and solve the following: 13. Find the indefinite integral and check the result by differentiation. Integral: x^2(x^3 - 1)^4 dx Answer: (x^3 - 1)^5/15 + C 130. Find the indefinite integral in two ways. Explain any difference in the forms of the answers. Integral: sin x cos x dx