Answered: Evaluate the integral ∫9−3f(x)dx ,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Evaluate the integral ∫9−3f(x)dx , where the graph of fest the red curve given in the following figure.
Please note that all important intersection points have integer coordinates. The blue line is for reference; it is not the x axis
.

-2
5
4
3
2
1
-1
-2
-3
2
X
Maple plot
9
8

Expert Solution

Step 1: Finding the function

The given semi circle has radius $3$ and center at $(6, 3)$

$f (x) = \{\begin{cases} \begin{matrix} x & f o r & - 3 \leq x \leq 3 \end{matrix} \\ 3 + \sqrt{9 - {(x - 6)}^{2}} f o r 3 \leq x \leq 9 \end{cases}$

$\int [9 - 3 f (x)] d x = \int_{- 3}^{3} (9 - 3 x) d x + \int_{3}^{9} 9 - 3 (3 + \sqrt{9 - {(x - 6)}^{2}}) d x = {[9 x - \frac{3 x^{2}}{2}]}_{- 3}^{3} + 3 \int_{3}^{9} \sqrt{9 - {(x - 6)}^{2}} d x = [27 - \frac{27}{2}] - [- 27 - \frac{27}{2}] + \frac{3}{2} {[x \sqrt{9 - {(x - 6)}^{2}} + 9 \sin^{- 1} (\frac{x - 6}{3})]}_{3}^{9} [∵ \int_{}^{} \sqrt{a^{2} - x^{2}} d x = \frac{x \sqrt{a^{2} - x^{2}}}{2} + \frac{a^{2} \sin^{- 1} (\frac{x}{a})}{2}] = 54 + \frac{3}{2} [0 + 9 \frac{π}{2}] - \frac{3}{2} [0 - 9 \frac{π}{2}] = 54 + \frac{27 π}{2}$