Calculate The Iterated Integral.103 12x3 9x2y2 Dy Dx

Calculating the iterated integral ∫∫ (103 + 12x³ + 9x²y²) dy dx involves integrating the function with respect to y first, then integrating the result with respect to x. This guide explains the process step-by-step and provides an online calculator to perform the calculation.

How to Calculate the Iterated Integral

The iterated integral ∫∫ (103 + 12x³ + 9x²y²) dy dx is calculated by first integrating with respect to y, then integrating the result with respect to x. This process is known as Fubini's theorem, which states that if the function is continuous over a rectangular region, the order of integration can be reversed.

Formula

∫∫ (103 + 12x³ + 9x²y²) dy dx = ∫ [∫ (103 + 12x³ + 9x²y²) dy] dx

To calculate this integral, you'll need to specify the limits of integration for both x and y. The calculator below allows you to input these limits and compute the result.

Step-by-Step Calculation

First Integration (with respect to y)

Integrate the integrand (103 + 12x³ + 9x²y²) with respect to y, treating x as a constant.

∫ (103 + 12x³ + 9x²y²) dy = 103y + 12x³y + 3x²y³ + C
Apply the Limits for y

Evaluate the antiderivative at the upper and lower limits for y.

[103y + 12x³y + 3x²y³] evaluated from y = a to y = b
Second Integration (with respect to x)

Integrate the result from the first integration with respect to x.

∫ [103(b - a) + 12x³(b - a) + 3x²(b³ - a³)] dx
Apply the Limits for x

Evaluate the antiderivative at the upper and lower limits for x.

[103(b - a)(x) + 4x⁴(b - a) + x³(b³ - a³)] evaluated from x = c to x = d

Worked Example

Let's calculate the iterated integral ∫∫ (103 + 12x³ + 9x²y²) dy dx over the region where x ranges from 0 to 1 and y ranges from 0 to 2.

Step 1: First Integration

∫ (103 + 12x³ + 9x²y²) dy = 103y + 12x³y + 3x²y³

Step 2: Apply y Limits (0 to 2)

[103(2) + 12x³(2) + 3x²(8)] - [103(0) + 12x³(0) + 3x²(0)] = 206 + 24x³ + 24x²

Step 3: Second Integration

∫ (206 + 24x³ + 24x²) dx = 206x + 6x⁴ + 8x³

Step 4: Apply x Limits (0 to 1)

[206(1) + 6(1)⁴ + 8(1)³] - [206(0) + 6(0)⁴ + 8(0)³] = 206 + 6 + 8 = 220

The result of the iterated integral is 220.

Interpreting the Result

The result of the iterated integral represents the volume under the surface defined by the function (103 + 12x³ + 9x²y²) over the specified region. In the example above, the volume is 220 cubic units.

Note: The result depends on the limits of integration. Different limits will produce different results.

Frequently Asked Questions

What is the difference between single and iterated integrals?

A single integral calculates the area under a curve, while an iterated integral calculates the volume under a surface. The iterated integral involves integrating with respect to one variable first, then the other.

When can I reverse the order of integration?

You can reverse the order of integration if the function is continuous over a rectangular region, as stated by Fubini's theorem.

How do I handle limits of integration in iterated integrals?

The limits of integration depend on the region of integration. For a rectangular region, you'll have separate limits for x and y.