Calculate The Iterated Integral Y Y2 Cos X Dx Dy

This guide explains how to calculate the iterated integral ∫∫ y y² cos(x) dx dy, including the step-by-step process, formula, and practical interpretation of results. The interactive calculator on this page makes it easy to compute the integral for any given limits.

What is an iterated integral?

An iterated integral is a double integral where one integral is evaluated first, followed by the second integral. For the integral ∫∫ y y² cos(x) dx dy, we first integrate with respect to x, then with respect to y.

Iterated integrals are used in physics, engineering, and mathematics to calculate quantities like work, probability, and mass distributions. The order of integration affects the result, so it's important to specify the limits of integration correctly.

Calculating ∫∫ y y² cos(x) dx dy

The integral ∫∫ y y² cos(x) dx dy represents the volume under the surface z = y y² cos(x) between the specified limits. To compute this, we'll use the following steps:

Integrate with respect to x first, treating y as a constant
Integrate the result with respect to y
Apply the limits of integration for both variables

Formula

∫∫ y y² cos(x) dx dy = ∫ [∫ y y² cos(x) dx] dy

Step-by-step solution

Let's solve ∫∫ y y² cos(x) dx dy from x=a to x=b and y=c to y=d.

Step 1: Integrate with respect to x

First, we integrate y y² cos(x) with respect to x, treating y as a constant:

∫ y y² cos(x) dx = y y² ∫ cos(x) dx = y y² sin(x) + C

Step 2: Apply the x limits

Now we apply the limits from x=a to x=b:

[y y² sin(x)] from a to b = y y² (sin(b) - sin(a))

Step 3: Integrate with respect to y

Next, we integrate the result with respect to y:

∫ y y² (sin(b) - sin(a)) dy = (sin(b) - sin(a)) ∫ y y² dy

To solve ∫ y y² dy, we can use substitution:

Let u = y², then du = 2y dy → dy = du/(2y)

∫ y y² dy = ∫ u (1/2) du = (1/2) (u²/2) + C = u²/4 + C = y⁴/4 + C

Step 4: Apply the y limits

Finally, we apply the limits from y=c to y=d:

(sin(b) - sin(a)) [y⁴/4] from c to d = (sin(b) - sin(a)) (d⁴/4 - c⁴/4)

Final result

∫∫ y y² cos(x) dx dy = (sin(b) - sin(a)) (d⁴ - c⁴)/4

Interpretation of results

The result (sin(b) - sin(a)) (d⁴ - c⁴)/4 represents the volume under the surface z = y y² cos(x) between the specified limits. The value can be positive or negative depending on the limits and the behavior of the sine function.

Example

For a=0, b=π/2, c=1, d=2:

sin(π/2) - sin(0) = 1 - 0 = 1

d⁴ - c⁴ = 16 - 1 = 15

Result = (1)(15)/4 = 3.75

Common mistakes to avoid

Incorrect order of integration - always integrate with respect to the innermost variable first
Forgetting to apply the limits of integration after each integration step
Miscounting the powers when integrating y y²
Incorrectly evaluating the sine function at the limits

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 order of integration matters in iterated integrals.

When would I use an iterated integral in real life?

Iterated integrals are used in physics to calculate work, in probability to find expected values, and in engineering to compute mass distributions.

How do I know which order to integrate first?

The order of integration depends on the limits of integration. For rectangular regions, the order doesn't matter, but for more complex regions, you may need to integrate with respect to the variable that has constant limits first.