Evaluate ∫∫ ඥ(4xଶ − yଶ) dxdy over the triangle formed bystraight lines y = 0, x = 1, y = x.

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The given problem is to evaluate the double integral of the function f(x, y) = 4x^2 - y^2 over the triangle formed by the lines y = 0, x = 1, and y = x.

Step 1: Identify the limits of integration The limits of integration are determined by the region over which the integral is being evaluated. In this case, the region is a triangle formed by the lines y = 0, x = 1, and y = x.

So, the limits of integration for y are 0 to x (from the line y = 0 to the line y = x), and the limits of integration for x are 0 to 1 (from the line x = 0 to the line x = 1).

Step 2: Set up the double integral The double integral is set up as follows:

∫ (from 0 to 1) [ ∫ (from 0 to x) (4x^2 - y^2) dy ] dx

Step 3: Evaluate the inner integral First, we integrate the function 4x^2 - y^2 with respect to y, treating x as a constant:

∫ (4x^2 - y^2) dy = 4x^2y - y^3/3, evaluated from y = 0 to y = x.

Substituting these limits of integration gives:

4x^3 - x^3/3 = 3x^3/3 - x^3/3 = 2x^3/3

Step 4: Evaluate the outer integral Next, we integrate the result 2x^3/3 with respect to x:

∫ (from 0 to 1) 2x^3/3 dx = x^4/2, evaluated from x = 0 to x = 1.

Substituting these limits of integration gives:

1/2 - 0 = 1/2

So, the value of the double integral ∫∫ (4x^2 - y^2) dxdy over the given triangle is 1/2.

The given problem is to evaluate the double integral of the function f(x, y) = 4x^2 - y^2 over the triangle formed by the lines y = 0, x = 1, and y = x.

Step 1: Identify the limits of integration The triangle is bounded by the lines y = 0, x = 1, and y = x. So, the limits of integration for y are 0 to x, and for x are 0 to 1.

Step 2: Set up the double integral The double integral is set up as follows: ∫ (from 0 to 1) ∫ (from 0 to x) (4x^2 - y^2) dy dx

Step 3: Evaluate the inner integral First, we integrate with respect to y: ∫ (from 0 to 1) [4x^2y - (y^3)/3] (from 0 to x) dx This simplifies to: ∫ (from 0 to 1) (4x^3 - x^3/3) dx

Step 4: Evaluate the outer integral Now, we integrate with respect to x: [4x^4/4 - x^4/12] (from 0 to 1) This simplifies to: 1 - 1/12 = 11/12

So, the value of the double integral over the given region is 11/12.

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