Double and Triple Integrals Calculator

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Example

Created on 2024-06-20Asked by Olivia Allen (Solvelet student)

Evaluate the double integral

\iint_R xy \, dA

, where

R

is the region bounded by

y = x

and

y = 2x

Solution

To evaluate the double integral

\iint_R xy \, dA

, where

R

is the region bounded by

y = x

and

y = 2x

:Step 1: Sketch the region

R

and identify the limits of integration. Step 2: Set up the double integral:

\iint_R xy \, dA = \int_{x=a}^{x=b} \int_{y=g(x)}^{y=h(x)} xy \, dy \, dx,

where

a

and

b

are the

x

-limits of integration, and

g(x)

and

h(x)

are the corresponding

y

-limits. Step 3: Determine the limits of integration by the intersection points of

y = x

and

y = 2x

x = y \quad \text{and} \quad x = 2y.

Solving these equations gives

x = 0

and

x = 0

as the limits of integration for

x

, and

y = 0

and

y = 2x

as the limits of integration for

y

. Step 4: Evaluate the integral:

\int_{0}^{1} \int_{0}^{2x} xy \, dy \, dx.

Step 5: Perform the integration:

\int_{0}^{1} \left[ \frac{1}{2}xy^2 \right]_{0}^{2x} \, dx = \int_{0}^{1} 2x^3 \, dx = \left[ \frac{1}{2}x^4 \right]_{0}^{1} = \frac{1}{2}.

Step 6: Conclusion. The value of the double integral

\iint_R xy \, dA

\frac{1}{2}

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Madison Ramirez on Solvelet

1. Evaluate the double integral

\int \int_{R} (x^2 + y^2) \, dA

, where

R

is the region bounded by the curves

y = x^2

and

y = x

.2. Use a triple integral to find the volume of the region bounded by the planes

x = 0

y = 0

z = 0

x + y + z = 1

, and

x + y + z = 2

DefinitionDouble and triple integrals are extensions of the idea of the integral to functions of two or three variables, respectively. They compute the signed volume under a topological surface or within a volume in n-dimensional space. Triple integrals are a generalization to integrate over regions (volumes, such as the cube inside the sphere) in 3-dimensional space (which are smooth 3-surfaces), while double integrals find the surface area of the well-defined regions. In areas of calculus, physics, engineering, and probability theory, double and triple integrals are methods used to determine the solutions to problems that relate to area, volume, mass, probability density, and multidimensional probability distributions. For instance, the double integral ∬Rf(x,y)dA means that it calculates the volume under the surface defined by the function f(x,y) when mapping over the region R in the xy-plane.

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