Calculate The Double Integral.6x1 Xyda R 0 6 0 1

This calculator computes the double integral of the function 6x1 xyda over the rectangular region R defined by x from 0 to 6 and y from 0 to 1. The result is presented in a clear, step-by-step format with visual aids where helpful.

Introduction

Double integrals extend the concept of single integrals to two dimensions. They are used to calculate quantities like area, volume, and average values over a two-dimensional region. The double integral of a function f(x,y) over a region R is written as:

∫∫_R f(x,y) dA

For rectangular regions, this can be evaluated using iterated integrals. This calculator handles the specific case of ∫∫_R 6x1 xyda over R = [0,6] × [0,1].

Double Integral Formula

The double integral over a rectangular region [a,b] × [c,d] is calculated using iterated integrals:

∫_a^b (∫_c^d f(x,y) dy) dx

For our specific problem:

∫₀⁶ (∫₀¹ 6x1 xyda) dx

This means we first integrate with respect to y from 0 to 1, then integrate the result with respect to x from 0 to 6.

Calculation Steps

First, integrate the integrand with respect to y from 0 to 1.
Then, integrate the resulting function with respect to x from 0 to 6.
Combine the results to get the final value of the double integral.

Note: The integrand 6x1 xyda is interpreted as 6x1 * x * y * da, where da represents the differential area element.

Worked Example

Let's compute ∫∫_R 6x1 xyda over R = [0,6] × [0,1]:

First, integrate with respect to y:
∫₀¹ 6x1 xy dy = 6x1 x ∫₀¹ y dy = 6x1 x [y²/2]₀¹ = 6x1 x (1/2 - 0) = 3x1 x
Now integrate the result with respect to x:
∫₀⁶ 3x1 x dx = 3x1 ∫₀⁶ x dx = 3x1 [x²/2]₀⁶ = 3x1 (18 - 0) = 54x1

The final value of the double integral is 54x1.

Frequently Asked Questions

What is the difference between single and double integrals?: Single integrals calculate quantities over a one-dimensional interval, while double integrals calculate quantities over a two-dimensional region.
When would I use a double integral in real life?: Double integrals are used in physics to calculate mass distributions, in engineering to find moments of inertia, and in probability to compute expected values over two-dimensional regions.
How do I know when to integrate with respect to x first or y first?: For rectangular regions, the order of integration doesn't affect the final result. However, the order can affect the complexity of the calculations.
What if my region isn't rectangular?: For non-rectangular regions, you may need to use more advanced techniques like polar coordinates or Green's theorem.