Calculate The Double Integral Lny Xy

The double integral of ln(y)xy is a fundamental concept in multivariable calculus. This guide explains how to compute it, provides an interactive calculator, and includes practical examples.

What is the double integral of ln(y)xy?

The double integral of ln(y)xy over a region D in the xy-plane is defined as:

∫∫_D ln(y)xy dA = ∫_a^b ∫_u(x)^v(x) ln(y)xy dy dx

This integral represents the volume under the surface z = ln(y)xy over the region D. It's commonly used in physics, engineering, and probability to calculate quantities like mass, charge, or probability density.

How to calculate the double integral of ln(y)xy

Step 1: Identify the region of integration

First, determine the region D over which you want to integrate. This could be a rectangle, a circle, or any other shape defined by boundaries.

Step 2: Set up the iterated integral

Express the double integral as an iterated integral. The order of integration (whether you integrate with respect to x first or y first) depends on the shape of the region D.

Step 3: Integrate with respect to the inner variable

First, integrate the integrand ln(y)xy with respect to the inner variable (either x or y).

Step 4: Integrate the result with respect to the outer variable

Take the result from the previous step and integrate it with respect to the outer variable.

Step 5: Evaluate the definite integral

Apply the limits of integration to the result from step 4 to obtain the final value.

Note: The exact steps may vary depending on the shape of the region D and the order of integration. Always double-check your setup before proceeding with the calculation.

Example calculation

Let's calculate the double integral of ln(y)xy over the region D defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ e^x.

Step 1: Set up the iterated integral

We'll integrate with respect to y first, then x:

∫₀¹ ∫₀^{e^x} ln(y)xy dy dx

Step 2: Integrate with respect to y

First, integrate ln(y)xy with respect to y:

∫ ln(y)xy dy = x ∫ ln(y)y dy

Let u = ln(y), dv = y dy. Then du = (1/y)dy, v = y²/2.

x ∫ ln(y)y dy = x [uv - ∫ v du] = x [y²/2 ln(y) - ∫ y²/2 (1/y) dy]

= x [y²/2 ln(y) - 1/2 ∫ y dy] = x [y²/2 ln(y) - y²/4]

Step 3: Integrate with respect to x

Now integrate the result from step 2 with respect to x:

∫₀¹ [x (y²/2 ln(y) - y²/4)] dx = (y²/2 ln(y) - y²/4) ∫₀¹ x dx

= (y²/2 ln(y) - y²/4) [x²/2]₀¹ = (y²/2 ln(y) - y²/4) (1/2 - 0) = y²/4 ln(y) - y²/8

Step 4: Evaluate the definite integral

Now evaluate the expression from step 3 at the limits y = e^x and y = 0:

[y²/4 ln(y) - y²/8]₀^{e^x} = [e^2x/4 x - e^2x/8] - [0 - 0]

= e^2x/4 x - e^2x/8

Step 5: Final integration with respect to x

Now integrate the result from step 4 with respect to x from 0 to 1:

∫₀¹ (e^2x/4 x - e^2x/8) dx = (1/4) ∫₀¹ x e^2x dx - (1/8) ∫₀¹ e^2x dx

For the first integral, use integration by parts with u = x, dv = e^2x dx:

∫ x e^2x dx = uv - ∫ v du = x e^2x/2 - ∫ e^2x/2 dx = x e^2x/2 - e^2x/4

Evaluating from 0 to 1:

[x e^2x/2 - e^2x/4]₀¹ = [e²/2 - e²/4] - [0 - e⁰/4] = e²/4 + 1/4

For the second integral:

∫ e^2x dx = e^2x/2

[e^2x/2]₀¹ = e²/2 - 1/2

Combining the results:

(1/4)(e²/4 + 1/4) - (1/8)(e²/2 - 1/2) = e²/16 + 1/16 - e²/16 + 1/16 = 1/8

Therefore, the value of the double integral is 1/8.

FAQ

What is the difference between single and double integrals?: A single integral calculates the area under a curve in two dimensions, while a double integral calculates the volume under a surface in three dimensions.
When would I use a double integral of ln(y)xy?: This integral is used in physics to calculate quantities like mass or charge distributions, in probability for probability density functions, and in engineering for various physical quantities.
How do I know which order of integration to use?: The order of integration depends on the shape of the region D. For simple regions like rectangles or circles, either order may work. For more complex regions, you may need to sketch the region to determine the correct order.
What if the integral doesn't converge?: If the integral doesn't converge, it means the quantity being calculated is infinite. This could indicate a problem with the setup or the region of integration.
Can I use this calculator for any region D?: Our calculator is designed for simple rectangular regions. For more complex regions, you may need to use a more advanced mathematical tool or software.