Calculate The Iterated Integral.203 4x3 18x2y2 Dy Dx

This calculator computes the double integral ∫∫ (203 + 4x³ + 18x²y²) dy dx over a specified rectangular region. The calculation follows standard integral evaluation techniques for polynomial functions.

How to Calculate the Iterated Integral

The iterated integral ∫∫ (203 + 4x³ + 18x²y²) dy dx is evaluated by first integrating with respect to y, then with respect to x over the given region. The process involves:

Identifying the limits of integration for x and y
Integrating the integrand with respect to y first
Integrating the resulting expression with respect to x
Evaluating the definite integral over the specified region

∫∫ (203 + 4x³ + 18x²y²) dy dx = ∫ [∫ (203 + 4x³ + 18x²y²) dy] dx = ∫ [203y + 4x³y + 6x²y³] dx = [203xy + x⁴y + 2x²y⁴] evaluated at bounds

The final result depends on the specific limits of integration you provide in the calculator.

Step-by-Step Calculation

Step 1: Set Up the Integral

Begin with the double integral expression:

∫∫ (203 + 4x³ + 18x²y²) dy dx

Step 2: Integrate with Respect to y

Integrate the integrand with respect to y first:

∫ (203 + 4x³ + 18x²y²) dy = 203y + 4x³y + 6x²y³ + C

Step 3: Integrate with Respect to x

Now integrate the result with respect to x:

∫ [203y + 4x³y + 6x²y³] dx = 203xy + x⁴y + 2x²y⁴ + C

Step 4: Apply the Limits of Integration

Evaluate the expression at the specified bounds for x and y to get the final result.

Worked Example

Let's calculate the integral from x=0 to x=1 and y=0 to y=1:

∫₀¹ ∫₀¹ (203 + 4x³ + 18x²y²) dy dx = ∫₀¹ [203y + 4x³y + 6x²y³]₀¹ dx = ∫₀¹ [203(1) + 4x³(1) + 6x²(1)³] dx = ∫₀¹ (203 + 4x³ + 6x²) dx = [203x + x⁴ + 2x³]₀¹ = 203 + 1 + 2 = 206

For this specific region, the integral evaluates to 206.

Frequently Asked Questions

What is the difference between single and double integrals?: A single integral calculates area under a curve in one dimension, while a double integral calculates volume under a surface in two dimensions.
How do I choose the order of integration?: The order of integration (dy dx or dx dy) depends on the region of integration. For rectangular regions, either order works.
Can I use this calculator for non-rectangular regions?: This calculator is designed for rectangular regions. For more complex regions, you would need to use polar or other coordinate systems.
What if my integrand has singularities?: If your integrand has singularities within the region, the integral may not converge. The calculator will show an error in such cases.