Chapter 4: Q30MP (page 240)

Evaluate $\frac{d^{2}}{{dx}^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) dsdt$ .

Short Answer

Expert verified

Thus, . $\frac{d^{2}}{d x^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) d s d t = 2 f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t + \int_{0}^{x} \frac{\partial f}{\partial x} d s$

Step by step solution

Determine the integral value.

$\frac{d^{2}}{d x^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) d s d t = \frac{d}{d x} [\frac{d}{d x} \int_{0}^{x} d s \int_{0}^{x} f (s, t) d t] = \frac{d}{d x} \int_{0}^{x} f (x, t) d t + \int_{0}^{x} d s \frac{\partial}{\partial x} \int_{0}^{x} f (s, t) d t = \frac{d}{d x} [\int_{0}^{x} f (x, t) d t + \int_{0}^{x} [f (s, x) + \int_{0}^{x} \frac{\partial f (s, t)}{\partial x} d t]]$

So, we get,

$\frac{d^{2}}{d x^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) d s d t = \frac{d}{d x} [\int_{0}^{x} f (x, t) d t + \int_{0}^{x} f (s, x) d s] = \frac{d}{d x} [\int_{0}^{x} f (x, t) d t] + \frac{d}{d x} [\int_{0}^{x} f (s, x) d s]$

Determine the Solution using formulas.

Now we use the formulas are,

$\frac{d}{d x} \int_{0}^{x} f (x, t) d t = f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t \frac{d}{d x} \int_{0}^{x} f (s, x) d s = f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d s$

Substituting these formulas,

$\frac{d^{2}}{d x^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) d s d t = f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t + f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d s = 2 f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t + \int_{0}^{x} \frac{\partial f}{\partial x} d s$

So, the solution is,

$\frac{d^{2}}{d x^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) d s d t = 2 f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t + \int_{0}^{x} \frac{\partial f}{\partial x} d s$

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 $\frac{d^{2}}{{dx}^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) dsdt$ .

Short Answer

Step by step solution

Determine the integral value.

Determine the Solution using formulas.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Space Physics

Fundamentals of Physics

Classical Mechanics

Particle Model of Matter

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Evaluate d2dx2∫0x∫0xfs,tdsdt .

Short Answer

Step by step solution

Determine the integral value.

Determine the Solution using formulas.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Space Physics

Fundamentals of Physics

Classical Mechanics

Particle Model of Matter

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Evaluate $\frac{d^{2}}{{dx}^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) dsdt$ .