Chapter 5: Q5P (page 247)

Evaluate the integrals $\int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x$ .

Short Answer

Expert verified

The value of given integral is $\int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x i s \frac{3 e^{2} - 5}{12}$ .

Step by step solution

Given data

The given equation is $\int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x$ .

Concept of Partial differential equation

Integration is a technique of finding a function $g (x)$ the derivative of which, $D g (x)$ , is equal to a given function $f (x)$ .

This is indicated by the integral sign " $\int$ , as in $\int f (x)$ , usually called the indefinite integral of the function.

Differentiate the equation

Consider the equation as follows:

$\int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x = \int_{x = 0}^{1} d x (\int_{y = x}^{e^{x}} y d y) \int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x = \int_{x = 0}^{1} d x {(\frac{y^{2}}{2})}_{y = x}^{e^{x}} \int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x = \frac{1}{2} \int_{x = 0}^{1} (e^{2 x} - x^{2}) d x \int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x = \frac{1}{2} {(\frac{e^{2 x}}{2} - \frac{x^{3}}{3})}_{0}^{1}$

Further, solve the equation as follows:

$\int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x = \frac{1}{4} (e^{2} - 1) - \frac{1}{6} \int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x = \frac{3 e^{2} - 5}{12}$

Therefore, the value of given integral is $\int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x = \frac{3 e^{2} - 5}{12}$ .

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.

Evaluate the integrals $\int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x$ .

Short Answer

Step by step solution

Given data

Concept of Partial differential equation

Differentiate the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Physical Quantities And Units

Electricity and Magnetism

Modern Physics

Mechanics and Materials

Measurements

Study anywhere. Anytime. Across all devices.

Evaluate the integrals ∫x=01∫y=xexydydx .

Short Answer

Step by step solution

Given data

Concept of Partial differential equation

Differentiate the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Physical Quantities And Units

Electricity and Magnetism

Modern Physics

Mechanics and Materials

Measurements

Study anywhere. Anytime. Across all devices.

Evaluate the integrals $\int_{x = 0}^{1} \int_{y = x}^{e^{x}} y d y d x$ .