Calculate The Iterated Integral 8 8 Π 2 Y Y2

This calculator computes the double iterated integral from 8 to 8, π to 2, and y to y². It shows the step-by-step solution and provides a visual representation of the integral.

What is this calculator?

The iterated integral calculator solves double integrals of the form ∫∫ f(x,y) dy dx over specified limits. This tool is useful for calculus students, engineers, and researchers working with multiple integrals.

Double integrals represent the volume under a surface or the area of a region in two dimensions. The calculator handles integrals of the form ∫ from a to b ∫ from c to d f(x,y) dy dx.

How to calculate the iterated integral

To compute the iterated integral ∫ from 8 to 8 ∫ from π to 2 y² dy dx:

First, solve the inner integral with respect to y from π to 2.
Then, integrate the result with respect to x from 8 to 8.
Simplify the final expression to get the value of the integral.

Note: The limits of integration must be constants or functions of the other variable. The integrand must be continuous over the region of integration.

Formula used

The general formula for the double iterated integral is:

∫ from a to b ∫ from c to d f(x,y) dy dx

For the specific case of ∫ from 8 to 8 ∫ from π to 2 y² dy dx:

First, integrate y² with respect to y from π to 2:

∫ from π to 2 y² dy = [y³/3] from π to 2 = (2³/3) - (π³/3) = (8/3) - (π³/3)

Then, integrate the result with respect to x from 8 to 8:

∫ from 8 to 8 [(8/3) - (π³/3)] dx = [(8/3) - (π³/3)] * (8 - 8) = 0

Worked example

Let's solve ∫ from 8 to 8 ∫ from π to 2 y² dy dx step-by-step:

First, compute the inner integral with respect to y:

∫ from π to 2 y² dy = [y³/3] from π to 2 = (2³/3) - (π³/3) = (8/3) - (π³/3)

Then, compute the outer integral with respect to x:

∫ from 8 to 8 [(8/3) - (π³/3)] dx = [(8/3) - (π³/3)] * (8 - 8) = 0

The final result is 0, which means the volume under the surface y² from x=8 to x=8 and y=π to y=2 is zero.

FAQ

What is the difference between single and double integrals?

Single integrals calculate area under a curve, while double integrals calculate volume under a surface or area of a region in two dimensions.

When would I use this calculator?

This calculator is useful for calculus students, engineers, and researchers working with multiple integrals in physics, engineering, and mathematics.

What if the integral doesn't converge?

If the integral doesn't converge, the calculator will indicate that the integral is divergent. This typically happens with improper integrals.