contestada

The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. y = −x^2 + 12x − 35,    y = 0;    about the x-axis

Respuesta :

up vote2down voteacceptedI think your limits of integration are incorrect. If you substitute y=4y=4 into y2−x2=9y2−x2=9, you find that x=±7–√x=±7. Therefore, the two curves intersect at x=±7–√x=±7. By washer method, we have:V=π∫7√−7√(4)2−(x2+9−−−−−√)2dx=2π∫7√016−(x2+9)dx=2π∫7√07−x2dx=2π[7x−13x3]7√0=2π(147–√3)=28π7–√3V=π∫−77(4)2−(x2+9)2dx=2π∫0716−(x2+9)dx=2π∫077−x2dx=2π[7x−13x3]07=2π(1473)=28π73And just for fun, let's try the shell method. Here, we have no choice but to find the volume obtained by revolving just the part of the region in the first quadrant, and doubling it.VVVVVV=2×2π∫43yy2−9−−−−−√dy=4π∫43yy2−9−−−−−√dy=4π[13(y2−9)32]43=4π[13(y2−9)32]43=4π[77–√3]=28π7–√3
azikennamdi

From the information given, the value of x is bounded within the region of X = -5, -7.



What is a Region bounded by curves?

To derive the above solution, we need to use the old fashioned quadratic method. The washer method cannot be used given that we were not given upper and lower limits.

Hence, since y is also equal to zero; we state:

−x² + 12x − 35 = 0; this can also be restated as:
x² + 12x + 35 = 0

To resolve this problem, we use the quadratic formula indicated below:

[tex]x = \frac{-b \pm \sqrt{b^{2} -4ac } }{2a}[/tex]

recall that the equation is:

x² + 12x + 35 = 0; where

a = 1

b = 12

c = 35

Hence,

[tex]x = \frac{-12 \pm \sqrt{12^{2} -4*1*35 } }{2*1}[/tex]

Step 2 - Simplify

[tex]x = \frac{-12 \pm \sqrt{{144} -4*1*35 } }{2*1}[/tex]

⇒ [tex]x = \frac{-12 \pm \sqrt{{144} -140 } }{2*1}[/tex]

⇒ [tex]x = \frac{-12 \pm 2}{2}[/tex]

Step 3 - Separate

x = -12 + 2/2

x = -12 - 2/2


Thus;

x = -12/2 = -5

x = -12-2/2 = -7

Hence, the boundaries of x are (-5, -7)

Learn more about area bounded by the given curve at;
https://brainly.com/question/15122151
#SPJ9

Ver imagen azikennamdi