Calculate The Double Integral 5 1 X2 1 Y2

This guide explains how to calculate double integrals of the form ∫∫ from 5 to 1 of x² dy dx and from 1 to y² dx dy. We provide a step-by-step method, an interactive calculator, and practical examples.

How to calculate double integrals

A double integral calculates the volume under a surface defined by z = f(x,y) over a region in the xy-plane. The general form is:

∫∫_R f(x,y) dA = ∫∫_R f(x,y) dx dy

For your specific integral ∫∫ from 5 to 1 of x² dy dx and from 1 to y² dx dy, we'll use the method of iterated integrals. Here's the step-by-step process:

Identify the region of integration. For your integral, the region is bounded by x = 1 to x = y² and y = 1 to y = 5.
Set up the iterated integral:
∫ from y=1 to y=5 [ ∫ from x=1 to x=y² x² dx ] dy
First, compute the inner integral with respect to x:
∫ x² dx = (x³)/3 evaluated from 1 to y²
Then compute the outer integral with respect to y:
∫ [(y⁴)/3 - 1/3] dy = (y⁵)/15 - y/3 evaluated from 1 to 5
Evaluate the definite integral:
[(5⁵)/15 - 5/3] - [(1⁵)/15 - 1/3] = (3125/15 - 5/3) - (1/15 - 1/3)

Note: The exact value of this integral is approximately 195.833. The calculator below will compute this for you automatically.

Example calculation

Let's work through a similar example to understand the process better. Calculate ∫∫ from 2 to 1 of x dy dx and from 1 to x dx dy.

Identify the region: x = 1 to x = 2, y = 1 to y = x
Set up the iterated integral:
∫ from x=1 to x=2 [ ∫ from y=1 to y=x 1 dy ] dx
Compute the inner integral:
∫ 1 dy = y evaluated from 1 to x = x - 1
Compute the outer integral:
∫ (x - 1) dx = (x²)/2 - x evaluated from 1 to 2
Evaluate the definite integral:
[(4/2 - 2) - (1/2 - 1)] = [2 - 2 - 0.5 + 1] = 0.5

The result is 0.5, which represents the area under the curve y = x from x = 1 to x = 2.

FAQ

What is the difference between single and double integrals?: A single integral calculates area under a curve in 2D space, while a double integral calculates volume under a surface in 3D space.
When would I use a double integral in real life?: Double integrals are used in physics for calculating mass distributions, in engineering for finding centroids, and in probability for calculating expected values over regions.
How do I know when to use dx dy vs dy dx?: The order of integration depends on the region of integration. For simple regions, either order may work, but for more complex regions, you'll need to adjust the limits accordingly.
What if my integral doesn't have simple limits?: For integrals with curved boundaries, you may need to use polar coordinates or other coordinate transformations to simplify the limits.