Chapter 5: Q30P (page 248)

$\int_{x = 0}^{2} \int_{y = x}^{2} e^{- y^{2} / 2} dydx$

Short Answer

Expert verified

The required solution is $1 - e^{- 2}$

Step by step solution

Definition of double integral

The double integral of f(x,y)over the area Ain the (x,y)plane as the limit of this sum, and we write it as

$\int \int_{A} f (x, y) dxdy$

Drawing the area bounded by the curve

The area of integration or bounded region is shown below,

Calculation of the area under the curve changing boundaries

Changing the boundaries of the given integration:

$y : 0 \to 2, and x : 0 \to y$

So, the integration

$l = \int_{0}^{2} e^{- y^{2} / 2} d y \int_{0}^{y} d x = \int_{0}^{2} e^{- y^{2} / 2} y d y = \int_{0}^{2} e^{- y^{2} / 2} d (- y^{2} / 2) = {e^{- y^{2} / 2}}_{0}^{2} = - (e^{- 2} - 1) = 1 - e^{- 2}$

Therefore, the value of the integration is, $1 - e^{- 2}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$\int_{x = 0}^{2} \int_{y = x}^{2} e^{- y^{2} / 2} dydx$

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Calculation of the area under the curve changing boundaries

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Fluids

Kinematics Physics

Geometrical and Physical Optics

Waves Physics

Space Physics

Study anywhere. Anytime. Across all devices.

∫x=02∫y=x2e-y2/2dydx

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Calculation of the area under the curve changing boundaries

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Fluids

Kinematics Physics

Geometrical and Physical Optics

Waves Physics

Space Physics

Study anywhere. Anytime. Across all devices.

$\int_{x = 0}^{2} \int_{y = x}^{2} e^{- y^{2} / 2} dydx$