Calculate The Double Integral Y 2 Y 1 Chegg

Double integrals are used to calculate areas, volumes, and other quantities in two-dimensional space. This guide explains how to compute double integrals with limits y2 and y1, including the setup, formula, and practical applications.

What is a Double Integral?

A double integral extends the concept of single integration to two dimensions. It calculates the volume under a surface defined by a function f(x,y) over a region in the xy-plane. The limits of integration, y2 and y1, define the vertical boundaries of the region.

Double integrals are essential in physics, engineering, and mathematics for solving problems involving density, temperature distribution, and fluid flow.

Double Integral Formula

∫∫_R f(x,y) dA = ∫_a^b [∫_y1(x)^y2(x) f(x,y) dy] dx

Where:

f(x,y) is the integrand function
y1(x) and y2(x) are the lower and upper vertical limits
a and b are the horizontal limits

The formula shows that a double integral is evaluated by first integrating with respect to y (from y1 to y2) and then integrating the result with respect to x (from a to b).

How to Calculate a Double Integral

Identify the region of integration R and determine the limits of integration.
Set up the double integral using the formula above.
Evaluate the inner integral (with respect to y) first.
Integrate the result with respect to x.
Evaluate the definite integral using the given limits.

Tip

For complex regions, it may be necessary to split the integral into simpler parts or use polar coordinates.

Worked Example

Calculate the double integral of f(x,y) = x²y over the region bounded by y = x² and y = x from x = 0 to x = 1.

Example Setup

∫₀¹ [∫_x²^x x²y dy] dx

First, evaluate the inner integral with respect to y:

∫ x²y dy = (x²/2)y² evaluated from y = x² to y = x

= (x²/2)x² - (x²/2)(x²) = (x⁴/2) - (x⁴/2) = 0

The result is 0, which makes sense because the region is symmetric and the function is odd in y.

FAQ

What is the difference between single and double integrals?

Single integrals calculate area under a curve in one dimension, while double integrals calculate volume under a surface in two dimensions.

When would I use a double integral?

Double integrals are used in physics for mass calculations, engineering for stress analysis, and mathematics for probability density functions.

How do I determine the limits of integration?

The limits are determined by the region of integration. For simple regions, sketch the region and identify the vertical and horizontal boundaries.