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Body Volume Rotation Of Shape Question

I want to know if Im following the correct step to evaluate the body volume rotation of shape. my function is : $$y=ln(x)$$ and I want to evaluate the body volume rotation of it between $y=0$ and ...
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Im trying to understand how should I evaluate this indefinite integral with this data on the integral : the question is : "Draw the shapes on the plain blocked - by the data lines and evaluate" : 1) ...
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The sum of the integration of g and $g^{-1}$

Let $g$ be a strictly increasing continuous function mapping $[a,b]$ onto $[A,B]$, and, as usual, let $g^{-1}: [A,B] \to [a,b]$ denote its inverse function. Use geometric insight to visualize the ...
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I'm really struggling to go about starting the following problem: This question concerns the integral, $\int_{0}^{2}\int_0^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ ... 2answers 38 views Integral of$\int^1_0\frac{dx}{\sqrt{x+3}-1}$I want to solve this integral and need some directions. $$\int^1_0\frac{dx}{\sqrt{x+3}-1}$$ I decided to call$x+3 = t^2 \rightarrow 2tdt = dx$then : $$\int^1_0 \frac {2tdt}{t^2-1}$$ Now what should ... 3answers 65 views The indefinite integral$\int \frac{1+\cos(x)}{\sin^2(x)}\,\mathrm dxI`m trying to solve this integral and I did the following steps to solve it but don't know how to continue. $$\int \frac{1+\cos(x)}{\sin^2(x)}\,\mathrm dx$$ \begin{align}\int \frac{\mathrm ... 2answers 31 views Integral of \int \frac{\sin(x)dx}{3-\cos(x)} I am trying to solve this integral and I need your suggestions. I don't know if its OK to set 3-\cos(x) as t \rightarrow dt = \sin(x)dx or just take \cos(x) and set it as t\int ... 4answers 70 views \frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$I'm having trouble understanding how to apply the$\frac{d}{dx}$when taking the anti-derivative. $$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$$ In class it was mentioned we'll end up taking ... 1answer 33 views Evaluating Complex Line Integrals Calculate$\int_{\gamma}\frac{\Re(z)}{z-\frac{1}{2}}dz$and$\int_{\gamma}\frac{\Im(z)}{z-\frac{1}{2}}dz$when$\gamma$:$|z|=1$is positively oriented. This is what I have tried to do, starting ... 0answers 43 views $\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)}$I'm stuck on my last exercise. Could you help? $$\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)} \ dx$$ 0answers 50 views Double Integral Homework Problem Here's the problem statement of the question which I am stuck on: Let$R_{1}$denote the rectangle$[0, 5] \times [-4, 4]$,$R_{2}$the rectangle$[0, 5] \times [0, 4]$, and$R_{3}$the rectangle ... 2answers 31 views Quadrature formula How can we find a quadrature formula$\int_{-1}^1 f(x) dx=c \displaystyle \sum_{i=0}^{2}f(x_i)$that is exact for all quadratic polynomials? Thanks for help. 4answers 59 views Integral of$\int \frac{x^4+2x+4}{x^4-1}dx$[duplicate] I am trying to solve this integral and I need your suggestions. $$\int \frac{x^4+2x+4}{x^4-1}dx$$ Thanks 3answers 88 views Integral of$\int^1_0 \frac{dx}{1+e^{2x}}$I am trying to solve this integral and I need your suggestions. I think about taking$1+e^{2x}$and setting it as$t$, but I don't know how to continue now. $$\int^1_0 \frac{dx}{1+e^{2x}}$$ Thanks! 2answers 39 views Integral of$\int(4-2x)^\frac{1}{3}dx$I solved this integral then I did$\frac{d}{dx}$of$F(x)$and saw that its not the same, so I did wrong in my integration process. $$\int(4-2x)^\frac{1}{3}dx$$ What I did is $$F(x) ... 5answers 85 views \int^1_0 \frac{xdx}{x^2+2x+1} I need some suggestion how to solve this integral.$$\int^1_0 \frac{xdx}{x^2+2x+1}$$I think about to do the following step :$$\frac{1}{2}\int^1_0\frac{2x+2-2dx}{x^2+2x+1} t=x^2+2x+1 \rightarrow ... 2answers 58 views Integral of fractional expression$\int^3_0 \frac{dx}{1+\sqrt{x+1}}$I want to solve this integral and think about call$\sqrt{x+1} = t \rightarrow t^2 = x+1$$$\int^3_0 \frac{dx}{1+\sqrt{x+1}}$$ Now the integral is : $$\int^3_0 \frac{2tdt}{1+t}$$ now I need your ... 2answers 72 views Integral of$ \int_{-1}^{1} \frac{x^4}{x^2+1}\,dx $Any suggestions how to solve it? by parts? $$\int_{-1}^{1} \frac{x^4}{x^2+1}dx$$ Thanks! 1answer 40 views real analysis: continous Let$g$be an increasing function on$[a,b]$to$\mathbb{R}$and suppose that for each$t ∈[c,d]$, the integral $$F(t) = \int_{a}^{b}f(x,t)\,dg(x)$$ exists Show that if$f_t$is continuous on ... 0answers 27 views Show derivative of integral equals integral of partial derivative if M[0,1]-measurable I am trying to determine a method of approaching the following: Suppose that$f:[0,1] \times (0,1)\rightarrow\mathbb{R}$is such that, for each$y \in (0,1)$, the function$f^{[y]}(x) = f(x,y)$... 0answers 43 views Interchange theorem for the Riemann-Stieltjes integral Let$J_1=[a,b]$and$J_2=[c,d]$. Assume that the real valued function$g$is monotone on$J_1$, that$h$is monotone on$J_2$, and that$f$is continuous on$J_1 \times J_2$. Define$G$on$J_2$and ... 1answer 68 views real analysis : futher properties of the integral Let$g$be an increasing function on$J_1 = [a,b]$to$\mathbb{R}$and for each fixed t in$J_2=[c,d]$, suppose that the integral $$F(t) = \int_{a}^{b}f(x,t)dg(x)$$ exists. If the partial ... 2answers 39 views Explain The Following Attribute Of Integral Explain The Following Attribute Of Integral: $$\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$$ I know that$ \int_a^a f(x)\,dx = 0 $but how it helps me? Thanks! EDIT Tought about ... 5answers 93 views Integral Of$\int \cos^4(x)dx$I want to solve this integral :$$\int \cos^4(x)dx$$ And think about doing the following thing:$\int (1-\sin^2(x))^2dx \rightarrow \int (1-2\sin^2(x)+\sin^4(x)dx$but I think I just complicated it. ... 1answer 79 views Can somebody provide an explanation to the formula of a one elementary integral? Here is the formula: $$\int{\frac{dx}{x}} = \ln{|x|} + C$$ In my textbook it is given without proof, so I have a little confusion here. From the definition of integral this equality must be true: ... 4answers 71 views Integral of$\int \sin(x) \cos(3x)dx$I want to solve this integral and I know that if I have non parity strong I can set$t=\cos(x) , t=\sin(x)$but what about the$\cos(3x)$I don't know now how to set$t$$$\int \sin(x) \cos(3x)dx$$ ... 1answer 50 views Laplace equation and integral $$\int_0^{2\pi} \frac{1+3 \sin{\phi}}{a^2-2ar \cos(\theta - \phi) + r^2 } d\phi$$ Help me plz ... I have tried to solve this. but I still don't know. 5answers 87 views Integral Of$\int\sqrt{\frac{x}{x+1}}dx$I want to solve this integral $$\int\sqrt{\frac{x}{x+1}}dx$$ And think about: 1)$t=\frac{x}{x+1}$2)$dt = (\frac{1}{x+1} - \frac{x}{(x+1)^2})dx$Now I need your advice! Thanks! 0answers 21 views Can someone help with this hydrostatic force question? [closed] A gate is in the form of a trapezoid 8 ft wide at the top, 5 ft wide at the bottom and is 3 ft high. If water reaches the top of the gate, what is the total hydrostatic force on the face of the gate? ... 0answers 39 views Can someone spot my error in the this question involving work done? A 2000 lb elevator is suspended by a 200 ft cable that weighs 10 lb/ft. How much work is done in lifting the elevator 40 feet? That cable starts out with 200 ft out. I start with this: 2000 lbs ... 1answer 42 views Can I get some assistance with this intregral / area problem? The problem states: Set up the integral needed to find the volume of the solid formed by revolving the area between$y = cosx$and$y = x, x = 0$around the$y$axis. The first thing I did was find ... 2answers 95 views Integral of$\frac{{x^{1/2}}+3}{2+{x^{1/3}}}$I want to solve this integral and think about doing the following steps:$1)\quad t=x^{1/3}2)\quad x=t^33)\quad dx=2t^3\,dt$How I can show$\sqrt{x}$as$t$? ... 3answers 69 views Integral Of$\frac{x^4+2x+4}{x^4-1}$Any ideas how to solve it? $$\int\frac{x^4+2x+4}{x^4-1}dx$$ Thanks! 2answers 46 views Integral of$\int\frac{(x^4+1)\,dx}{x^3+4x}$I followed the steps to solve this integral and want to know if I did it right and if$C=0? $$$\int\frac{(x^4+1)\,dx}{x^3+4x} = \int\frac{(x^4+1)\,dx}{x(x^2+4)} = \frac{A}{x}+\frac{Bx+C}{x^2+4}$$ ... 2answers 37 views Is there a need for another integration technique? I'm being asked to calculate $$I\triangleq\int_0^1\int_{e^{\large x}}^e{xe^y\over(\ln y)^2}\,dy\,dx\quad.$$ I got stuck on the indefinite inner one, $$J\triangleq\int{e^ydy\over(\ln y)^2}\quad.$$ At ... 2answers 42 views Work done by a force Field Homework for Calc III includes a problem about computing the work done by a force field (defined by a specific vector equation) on a moving particle. I was attempting to compute this using the ... 1answer 38 views Question About the Integration of rational function I was asked to dismantle this rational function by parts and wanted to know if I did it right. The function is: $${\frac{x^5-x+3}{x(x-2)^3(x^2+2x+2)}}$$ What I did is: ... 1answer 27 views Finding volume under surface and above a region I'm asked to find$\underset{U}{\int}(x+y)^2\, dA$where U is a region bounded by the lines x = -1, x = 1, y = -1 ... and by the curves x=$y^2$, y=1+$x^2$Plot: http://d.pr/WYSg I started out by ... 1answer 45 views Finding area between two polar curves using double integrals I have a homework question that is asking me to find the area that lies: Inside the curve$r=2+cos(2\theta)$But outside the curve$r=2+sin(\theta)$I think I'm supposed to be using a double ... 1answer 34 views Integral question -$\int\frac{(x+6)\,dx}{4x-x^2}$Integral question -$\int\frac{(x+6)\,dx}{4x-x^2}$What I did is $$\int\frac{(x+6)\,dx}{x(4-x)}$$ then $$\int\frac{(x+6)\,dx}{4x-x^2}= \int\left(\frac{A \,}{x}+\frac{B}{4-x}\right) dx$$ this is the ... 2answers 33 views Integral question -$\int\frac{(4-x)\,dx}{x^2+4x+8}$Integral question - $$\int\frac{(4-x)\,dx}{x^2+4x+8}$$ To solve it I need to bring the numerator to be the derivative of the dominator right? I need to do the trick that not change the integral any ... 3answers 59 views Integral question -$\int\frac{\sqrt{\tan(x)}}{{\cos^2(x)}}dx$Integral question - $$\int\frac{\sqrt{\tan(x)}}{{\cos^2(x)}}dx$$ I see that$\frac{1}{\cos^2(x)}$is the derivative of$\tan(x)$so I set$t = \tan(x)$? or the whole square? Thanks! 1answer 34 views Integral question -$\int\frac{\sin\sqrt{x}}{\sqrt{x}}$This is the integral : $$\int\frac{\sin\sqrt{x}}{\sqrt{x}}$$ I thinking about put universal identity but not sure, I know that$\sin(x) = \dfrac{2t}{1+t^2}$. But what about the square root? Instead ... 6answers 166 views Integral Question -$\int\frac{1}{x^2-6x}\,\mathrm dx$How I can solve it? : $$\int\frac{1}{x^2-6x}\,\mathrm dx$$ Do I need to bring it to this format? :$\displaystyle \int\frac{1}{x^2-a^2}\,\mathrm dx$? Thanks! 2answers 74 views Integral Question$\int\frac{\sin^4(x)}{\cos^2(x)}\,dx$What you are suggesting to do? Convert$\sin^4(x)\Rightarrow (1-\cos^2(x))^2\,dx?$$$∫\frac{\sin^4(x)}{\cos^2(x)}\,dx$$ Thanks! 2answers 28 views Is it true that$\int_1^ba^{\log_b x}dx> \log_eb$Is it true that$\int_1^ba^{\log_b x}dx> \log_eb\forall a,b>0\ and\ b\not = 1$3answers 39 views Integrals with variables on top and bottom How do I solve an integral that has variables on both top and bottom? To solve an integral like$\int^{x}_0t^2\$+5 dt, I would simply plug in x for t, and if both the top and bottom of an integral ...
$$F(x)=\int^{x}_{0} \frac{t^2-16}{1+\cos^2 t}\,dt.$$ The problem says to find the local max of this expression. AFAIK, to take the max or min, I have to take the derivative of that expression. To do ...