Double integral Definition and 558 Threads

Homework Statement >Problem:<br>Find the Moment of Inertia of a circular disk of uniform density about an axis which passes through the center and makes an angle of $\dfrac{\pi}{6}$ with the plane of the disc. Homework Equations Moment of Inertia ($I$) is $$\int r^2dm$$ where $r$ is the...

Very simple question for you, friends. As is well known, usual integral has interpretation as square under function's graphic. Then, what is double (and triple) integral by analogue? Thanks!

I was reading about double integral when a doubt came to my mind: how to find the antiderivative of the function f(x,y), like bellow, and compute the fundamental theorem of calculus for double integral? \int_{2}^{8} \int_{2}^{6} f(x,y) dx \wedge dy = ? OBS: It's not an exercise. I know how...

Homework Statement 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 to show that the magnitude of the force of attraction that the lamina exerts on a body of mass m located at the point (0,0,d) on the positive...

Homework Statement Let R be the triangle defined by -xtanα≤y≤xtanα and x≤1 where α is an acute angle sketch the triangle and calculate ∫∫R (x2+y2)dA using polar coordinates hint: the substitution u=tanθ may help you evaluate the integral Homework EquationsThe Attempt at a Solution so the...

I have what I think is a valid solution, but I'm not sure, and when I try to check the answer approximately in Matlab, I don't get a verified value, and I'm not sure if my analytic solution or my approximation method in Matlab is at fault. 1. Homework Statement Evaluate the integral...

I'm on a tablet and having trouble with the math symbols so, for clarity, ∫[a,b] xdx is the integral from a to b of x with respect to x, and f(x) |[a,b] is a function of x evaluated from a to b. Problem: ∫[-1,1]∫[-√(1 - y2),√(1 - y2)] ln(x2 + y2 + 1) Relevent Equations: x2 + y2 = r2 ∫udv =...

Hi, I need to evaluate the following double integral. I have tried direct integration but the answer is too complicated for it to be a viable method. First integral is from 0 to (1-y^2) function is (x^2+y^2)dx. Second integral is from 0 to 1 dy. I can't figure out how to use the maths thing...

Homework Statement Find the volume of the solid. Under the paraboloid z = x^2 + y^2 and above the region bounded by y = x^2 and x = y^2 Well, those curves only intersects in the xy-plane at (0,0) and (1,1), and in the first Quadrant, and in that first Quadrant y = sqrt(x), and over that...

Homework Statement \iint\limits_D x{\rm{d}}x{\rm{d}}y where x = \sqrt{2y - y^2}, y = \sqrt{2x - x^2} Homework EquationsThe Attempt at a Solution I have figured out the region in question: But how do I get the limits of integration? Ah, perhaps.. \int_0^1 \int_{1-\sqrt{1-y^2}}^{\sqrt{2y-y^2}}...

Hello, I was helping my friend prepare for a calculus exam today - more or less acting as a tutor. He had the following question on his exam review: ∫∫R y2 dA Where R is bounded by the lines x = 2, y = 2x + 4, y = -x - 2I explained to him that R is a triangle formed by all three of those...

Homework Statement Let ## E ## be the ellipsoid: $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$ Let ## S ## be the part of the surface of ## E ## defined by: $$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$ Let F be the vector field defined by $$ F=(-y,x,0)$$ A) Explain why ##...

Homework Statement Find the volume: Prism formed by x+z=1, x-z=1, y=2, y=-2, and the yz-plane. Homework EquationsThe Attempt at a Solution Okay, so I sketched the drawing and I found that I could take the upper region of the xy-plane with respects to x and z and a triangle was formed. The...

Homework Statement The region between sphere x^2+y^2+z^2=3 and the upper sheet of the hyperboloid z^2=x^2+y2+1. Homework EquationsThe Attempt at a Solution Curve of intersection: We set the two equations equal to each other and find x^2+y^2=1, a circle of radius 1 is the curve of...

Homework Statement Double integral of y*e^(x^4-1) with bounds 0=<y=<1 y^(2/3)=<x=<1Homework EquationsThe Attempt at a Solution [/B] Well, the first key thing to recognize is that we need the correct order for the bounds to compute this double integral. So I switch it from x=y^(2/3) and x=1 TO...

Homework Statement A thermally conducting, uniform and homogeneous bar of length L, cross section A, density p and specific heat at constant pressure cp is brought to a nonuniform temperature distribution by contact at one end with a hot reservoir at a temperature TH and at the other end with a...

Homework Statement Evaluate the double integral (x+2y)dA, where R is the region in the first quadrant bounded by the circle x^2+y^2=9. Homework Equations None. The Attempt at a Solution I know how to evaluate the double integral but I just don't know how to find the limits of integration. I...

Hello, I would like to differentiate the following expected value function with respect to parameter $$\beta$$: $$F(\xi_1,\xi_2) =\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\frac{\xi_1+\xi_2-2bK}{2(1-\beta)^2} g(\xi_1,\xi_2)d\xi_1 d\xi_2$$ $$g(\xi_1,\xi_2)$$ is...

This is the equation for the cone A \sqrt{x^2 + y^2} The double integral \iint A \sqrt{x^2 + y^2} \space dy \space dx \space \space \space\text {From x= -1 to 1 and y=} -\sqrt{1-x^2} \space to \space \sqrt{1-x^2} \text{ is very difficult to evaluate. I've tried polar coordinate substitution...

Homework Statement Please refer to : http://math.stackexchange.com/questions/1068948/how-to-prove-that-int-0-infty-sinx-arctan-frac1x-mathrm-dx-fra/1069065#1069065 The answer by @venus. What is the procedure in converting that single integral, dividing it into parts, and making it a double...

Hi! I'm stuck with the following problem: ----------------------------------- Calculate ∫∫ (x-y)*|ln(x+2y)| dxdy where D is the triangle with corners in the coordinates (0,0), (1,1) and (-3,3) ----------------------------------- I get the following lines that forms the triangle: y=-x, y=x...

how do i numerically calculate a double integral? as i understand simpsons 3/8 rule is the optimal method for a single integral, is it still true for double integrals? if so, how do i extend the 3/8s rule to do a double integral?

Homework Statement find the volume of the solid below the plane z = 4x and above the circle x^2 + y^2 = 16 in the xy plane Homework EquationsThe Attempt at a Solution This totally confused me. I didn't think the plane z = 4x sat above the xy plane. If that is true then there would be no solid...

Homework Statement Evaluate: I(y)= \int^{\frac{\pi}{2}}_{0} \frac{1}{y+cos(x)} \ dx if y > 1 Homework EquationsThe Attempt at a Solution I've never seen an integral like this before. I can see it has the form: \int^{a}_{b} f(x,y) dx I clearly can't treat it as one half of an exact...

Homework Statement Evaluate the integral by changing into polar coordinates. \displaystyle \int_0^{4a} \int_{y^2/4a}^y \dfrac{x^2-y^2}{x^2+y^2} dx dy The Attempt at a Solution Substituting x=rcos theta and y=rsin theta , the integrand changes to cos 2 \theta r dr d \theta . I know that the...

I can't compute the integral: \int \frac{\arccos(\sqrt{x^2+y^2})}{\sqrt{x^2+y^2}}\frac{x-a}/{(\sqrt{(x-1)^2+y^2})^3 dxdy on an unit circle: r < 1. for const: a = 0.01, 0.02, ect. up to 1 or 2. I used a polar coordinates, but the values jump dramatically in some places (around the 'a' values)...

Homework Statement Here is a more interesting problem to consider. We want to evaluate the improper integral \intop_{0}^{\infty}\frac{\tan^{-1}(6x)-\tan^{-1}(2x)}{x}dx Do it by rewriting the numerator of the integrand as \intop_{f(x)}^{g(x)}h(y)dy for appropriate f, g, h and then reversing...

Homework Statement evaluate the double integral of cosh(6x^2+10y^2) dxdy by making the change of variables x=rcos(theta)/sqrt(3) and y=rsin(theta)/sqrt(5) let D be the region enclosed by the ellipse 3x^2+5y^2=1 and the line x=0 where x>0. Homework EquationsThe Attempt at a Solution first I...

Homework Statement Find the volume under the surface z = y(x+2) and over the area bounded by y+x = 1, y = 1 and y = sqrt(x) Homework Equations The Attempt at a Solution Based on the geometry of the bounds, I broke this integral into two parts. I first found the intersection of...

Homework Statement This isn't actually homework. I was messing around in my notebook trying something when I ended up writing something to the effect of this: dT = \frac{V^{2}}{R(1+α dT)}dQ R(1+α dT) dT = V^{2}dQ Where α and V are constants. Now, I'm fairly sure what I had done made...

Homework Statement ##\mathscr{C}## is an ellipse ##\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1## and ##\vec{F}(x,y) = <xy^2, yx^2>## write ##\displaystyle \int_\mathscr{C} \vec{F} \cdot d\vec{s}## as a double integral using greens theorem and evaluate Homework Equations ##\displaystyle...

Homework Statement f(x,y) = ##e^{x+y}## D is the triangle vertices (0,0), (0,1) , (1,0) Homework Equations ##f(x,y)_{avg}=\frac{\iint_D f(x,y) dA}{\iint_D dA}## The Attempt at a Solution ##\iint_D dA \Rightarrow \int_{0}^{1}\int_{0}^{-y+1} dxdy = \frac{1}{2}## ##\iint_D...

Take any given point on the perimeter of a (A x B) rectange and then draw a line from that point to another point on one of the three remaining sides of the rectangle. What is the average length of the line? Well, the answer to that question involves integrals like this: \int_0^A \int_0^B...

Homework Statement ∫∫D√(9x2+4y2) dx dyD is the region: x2/4+y2/9=1 My understanding is that i have to integrate the function of a density to calculate the mass of plate which is ellipse. Problem is i can't and shouldn't be able to integrate this integral at my level, so am i missing some way...

Homework Statement Use the change of variables ##u=x+y## and ##y=uv## to solve: \int_0^1\int_0^{1-x}e^{\frac{y}{x+y}}dydx Homework Equations The Attempt at a Solution So I got as far as: \int\int{}ue^vdvdu. But I just can't find the region of integration in terms of ##u## and ##v##.

I'm trying to figure out what this one symbol was I saw. I also have a guess that I would like to see if is correct. I saw a double integral with a circle connecting the two. What does this mean? Here is my guess. Is it used when dealing with Stoke's Theorem? Since ∫F°dS =∫∫ curl(F)°dS (Both...

Hi I'm doing a surface area problem with a parametric surface and I got the cross product but I can't figure out the double integral. I found the solution online but with no explanation, so can someone explain how to solve this integral: thank you!

Homework Statement Homework Equations The Attempt at a Solution As with my other recent posts, I just want to check if I'm right or wrong as I don't have an answer scheme to go by. For this question I simply converted to polar to get: ∫∫(a+a)r drdθ for 0<r<a, 0<θ<2π ...

Homework Statement Use a double integral to find the volume of the indicated solid. Homework Equations The Attempt at a Solution I can't find what I did wrong, it seems like a simple problem... $$\int_0^2 \int_0^x (4-y^{2})dydx=\int_0^2 4x-\frac{x^{3}}{3}dx$$...

Homework Statement ∫∫e^(y√x)dxdy from 1 to 4 then from 0 to 2 Homework Equations ∫ e^x = e^x u substitution The Attempt at a Solution I am just curious if this is equal to double integral e^(y\sqrt{x})dydx from 0 to 2 then from 1 to 4. In other words can I change the order of...

Homework Statement Homework Equations The Attempt at a Solution For part B, why is he using the formula for the moment of inertia about the y-axis? Why isn't he using the formula for the moment of inertia about the origin...

The popular fundamental theorem of calculus states that \int_{x_0}^{x_1} \frac{df}{dx}(x)dx = f(x_1)-f(x_0) But I never see this theorem for a dobule integral... The FTC for a univariate function, y'=f'(x), computes the area between f'(x) and the x-axis, delimited by (x0, x1), but given a...

Hey. Homework Statement ∫∫x^3 dxdy, with the area of integration: D={(x,y)∈R^2: 1<=x^2+9y^2<=9, x>=3y} The Attempt at a Solution Did the variable substitution u=x and v=3y so the area of integration became 1<=u^2 + v^2 <=9, u>=v. And the integral became ∫∫(1/3)u^3 dudv. Then I switched to...

Homework Statement Evaluate \iint\limits_S \vec{A} . \vec{n} ds over the plane x^{2}+y^{2}=16, where \vec{A}=z\vec{i}+x\vec{j}-3y^{2}\vec{k} and S is a part from the plane and R was projected over xz-plane. Homework Equations Surface Integral and Double Integration.The Attempt at a...

Hi everyone, I've the equation x+y=6 (it's a surface equation which I'll integrate over) and the following integral limits is what I suppose to get it from the equation: \int\limits_0^6 \int\limits_0^{6-x} What's the trick here?

Can anybody please help me understand the computation of the integral in the attached image. I shall be grateful.

Homework Statement ∫∫ydxdy over the triangle with vertices (-1,0), (0,2), (2,0) Homework Equations I did it like this and got the right answer: ∫dy ∫ydx this first: ∫ydx from x = (y-2)/2 to x = 2-y then ∫dy from y = 0 to y = 2 I got 2 which is correct but when I...

Find the double integral of (integral sign) (integral sign) ydA where D is the region bounded by (x+1)^2, x=y-y^3, x=-1, and y=-1

Homework Statement ∫∫√(x^2+y^2)dxdy with 0<=y<=1 and -SQRT(y-y^2)<=x<=0 Homework Equations x=rcos(theta) y=rsin(theta) The Attempt at a Solution 0.5<=r=1, we get r=0.5 from -SQRT(y-y^2)<=x by completing the square on the LHS then, 0<=theta<=pi But, when I calculated the...

The problem is as follows. Calculate the double integral of cos ((x-y)/(x+y)) dA over R, where R is the triangle bounded by the points (0,0), (2,2), and (2 + pi, 2 - pi). I understand that you have to set U = x-y and V = x+y. However, I am having a hard time finding the bounds on the...