What is Double integral: Definition and 573 Discussions

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in

R

2

{\displaystyle \mathbb {R} ^{2}}
(the real-number plane) are called double integrals, and integrals of a function of three variables over a region in

R

3

{\displaystyle \mathbb {R} ^{3}}
(real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

View More On Wikipedia.org

3. Symmetry of an Integral of a Dot product

This homework statement comes from a research paper that was published in SPIE Optical Engineering. The integral $$\int\int_{-\infty}^{\infty}drdr'W(\vec{r})W(\vec{r'}) \vec{r} \cdot \vec{r'}=0$$ is an assumtion that is made via the following statement from the paper : "Since...
4. Two ways of integration giving different results

I am trying to do the double integral. And I remembered there's this formula that says if the integrand can be split into products of F(x) and G(y) then we can do each one separately, then take the product of each result. Taken from Stewart's Calculus 9E. So I tried to do the integral two...

46. S

Area Calculation for Circle and Cardioid Using Double Integrals

Homework Statement r=1 and r=1+cos(theta), use a double integral to find the area inside the circle r=1 and outside the cardioid r=1+cos(theta) Homework EquationsThe Attempt at a Solution I am confused on the wording and how to set it up. I tried setting it up by setting theta 0 to pi. and r...
47. MHB Double integral Problem (with solution)

Evaluate (use attached figure for depiction) $\iint_{R} \, xy \, dA$ where $R$ is the region bounded by the line $y = x - 1$ and the parabola $y^2 = 2 x + 6$. I will post solution in just a moment with a reply.
48. M

Using Green's Theorem for a quadrilateral

Homework Statement Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful. Homework Equations Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given...
49. S

Volume of Double Integral: Finding the Region with Graphed Equations

Homework Statement z=x^2+xy ,y=3x-x^2,y=x find the volume of the region Homework EquationsThe Attempt at a Solution I graphed y=3x-x^2 and y=x I am confused on which region I use to find the volume. Do I use the upper region or the lower region.
50. Hard Double Integral Homework: Solve & Understand

Homework Statement I'm given the integral show in the adjunct picture, in the same one there is my attempt at a solution. Homework Equations x = r.cos(Θ) y = r.sin(Θ) dA = r.dr.dΘ The Attempt at a Solution [/B] I tried to do it in polar coordinates, so I substituted x=r.cos(Θ) y=r.sin(Θ) in...