Calculate The Double Integral 9xsin X Y Da

Calculating the double integral of 9x sin(x) y da involves integrating the function 9x sin(x) y over a specified region in the xy-plane. This guide explains the process, provides a calculator, and offers practical examples.

How to Calculate the Double Integral

The double integral of a function f(x,y) over a region R in the xy-plane is calculated by integrating the function with respect to x first, then integrating the result with respect to y.

Double Integral Formula

∫∫_R f(x,y) dA = ∫_a^b [∫_u(x)^v(x) f(x,y) dy] dx

For the function 9x sin(x) y, we'll need to determine the limits of integration based on the region R. Common regions include rectangles, triangles, or other simple shapes.

Note: The exact limits of integration depend on the specific region R. The calculator below allows you to specify these limits.

The Formula

The double integral of 9x sin(x) y over a region R is calculated using the following steps:

Determine the limits of integration for x and y based on the region R.
Integrate 9x sin(x) y with respect to y first, using the limits for y.
Integrate the result with respect to x, using the limits for x.

Step-by-Step Calculation

First integral: ∫ 9x sin(x) y dy = 9x sin(x) ∫ y dy = 9x sin(x) (y²/2) evaluated from y=u(x) to y=v(x)
Second integral: ∫ [9x sin(x) (v(x)² - u(x)²)/2] dx from x=a to x=b

Worked Example

Let's calculate the double integral of 9x sin(x) y over the rectangle R defined by 0 ≤ x ≤ π and 0 ≤ y ≤ 1.

Example Calculation

First integral: ∫₀¹ 9x sin(x) y dy = 9x sin(x) [y²/2]₀¹ = 9x sin(x) (1/2 - 0) = (9/2)x sin(x)
Second integral: ∫₀^π (9/2)x sin(x) dx
Using integration by parts: ∫ x sin(x) dx = -x cos(x) + sin(x)
Final result: (9/2) [-x cos(x) + sin(x)]₀^π = (9/2) [-(π)(-1) + 0 - (0 + 0)] = (9/2)(π) = 4.7124

The result is approximately 4.7124. This represents the volume under the surface z = 9x sin(x) y over the specified rectangle.

Interpreting the Result

The double integral represents the volume under the surface defined by the function 9x sin(x) y over the region R. The result is a scalar value representing this volume.

For practical applications, the interpretation depends on the context. In physics, it might represent mass or charge density. In engineering, it could represent work or energy.

FAQ

What is a double integral?: A double integral extends the concept of single integration to two dimensions, calculating the volume under a surface over a region in the xy-plane.
When would I use a double integral?: Double integrals are used in physics for mass/charge density, engineering for work/energy, and probability for joint probability density functions.
How do I choose the limits of integration?: The limits depend on the region R. For simple shapes like rectangles or triangles, you can determine them from the region's boundaries.
What if the integral is difficult to compute?: For complex integrals, numerical methods or symbolic computation software can be used. Our calculator provides an approximate solution.
Can I calculate triple integrals with this tool?: This calculator is specifically for double integrals. For triple integrals, you would need a different tool or approach.