The function e is an example of a Gaussian (bell curve)'. A very useful result in probability/statistics is that dr = VT. In this problem, we'll use a double integral in polar coordinates to compute this integral. Let f(x, y) = e-¤²-y´ and let DR be the disk of radius R centered at the origin, x² + y² < R². Compute /I f(x, y) dA. Let R → o in your answer to part (a). The result is dA. 8. Use the result of part (b), together with the result of #2(a), to show that -72 dx VT. (Hint: the name of the variable in an integral does not matter.)
The function e is an example of a Gaussian (bell curve)'. A very useful result in probability/statistics is that dr = VT. In this problem, we'll use a double integral in polar coordinates to compute this integral. Let f(x, y) = e-¤²-y´ and let DR be the disk of radius R centered at the origin, x² + y² < R². Compute /I f(x, y) dA. Let R → o in your answer to part (a). The result is dA. 8. Use the result of part (b), together with the result of #2(a), to show that -72 dx VT. (Hint: the name of the variable in an integral does not matter.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The function \( e^{-x^2} \) is an example of a Gaussian (bell curve)\(^1\). A very useful result in probability/statistics is that
\[
\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.
\]
In this problem, we'll use a double integral in polar coordinates to compute this integral.
Let \( f(x, y) = e^{-x^2-y^2} \) and let \( D_R \) be the disk of radius \( R \) centered at the origin, \( x^2 + y^2 \leq R^2 \). Compute \(\int\int_{D_R} f(x, y) \, dA\).
Let \( R \to \infty \) in your answer to part (a). The result is
\[
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dA.
\]
Use the result of part (b), together with the result of \(\#2(a)\), to show that
\[
\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.
\]
(Hint: the name of the variable in an integral does not matter.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66e8a441-1535-4b77-8170-ec0b022e4104%2Fdeb28eb7-82fe-4847-96f8-adcc9b7ff3bb%2Fz3xo1k_processed.png&w=3840&q=75)
Transcribed Image Text:The function \( e^{-x^2} \) is an example of a Gaussian (bell curve)\(^1\). A very useful result in probability/statistics is that
\[
\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.
\]
In this problem, we'll use a double integral in polar coordinates to compute this integral.
Let \( f(x, y) = e^{-x^2-y^2} \) and let \( D_R \) be the disk of radius \( R \) centered at the origin, \( x^2 + y^2 \leq R^2 \). Compute \(\int\int_{D_R} f(x, y) \, dA\).
Let \( R \to \infty \) in your answer to part (a). The result is
\[
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dA.
\]
Use the result of part (b), together with the result of \(\#2(a)\), to show that
\[
\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.
\]
(Hint: the name of the variable in an integral does not matter.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
