If f is a constant function, f(x, y)=k, and R=[a, b] ×[c, d], show that ∬R k d A=k(b-a)(d-c)

Name: Problem 17, Chapter 15: Calculus Early Transcendentals
Uploaded: 2020-05-21T03:26:49-08:00
Duration: 3 min 1 s
Channel: Christian Otero
Description: If f is a constant function, f(x, y)=k, and R=[a, b] ×[c, d], show that ∬R k d A=k(b-a)(d-c)

step 1 All right, so what we're tasked with here is to prove this theorem, showing that the integral of

If f is a constant function, f(x, y)=k, and R=[a, b] ×[c,...

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Constant Function

A constant function is one that assigns the same value to every point in its domain. When integrating a constant function over any region, the result is simply the constant multiplied by the measure (or area) of that region.

Double Integration

Double integration is the process of integrating a function over a two-dimensional domain. It extends single-variable integration by summing values over regions in the plane, and it is used to calculate quantities like area and volume.

Area of a Rectangle

The area of a rectangle, defined as the product of the lengths of its sides, is crucial when integrating over a rectangular region. For the rectangle [a, b] × [c, d], the area is calculated as (b - a)(d - c), which directly scales the value of a constant integrand.

Iterated Integral and Fubini's Theorem

Fubini's Theorem allows the computation of a double integral as an iterated integral, meaning that one can perform the integration one variable at a time. This approach simplifies calculations over rectangular regions by converting a double integral into the product of two single integrals.

Transcript

Please give Ace some feedback