With the curve
![C](https://tex.z-dn.net/?f=C)
parameterized by
![C:\mathbf r(t)=\underbrace{15\sin t}_{x(t)}\,\mathbf i+\underbrace{13\sin^2t}_{y(t)}\,\mathbf j+\underbrace{12\sin^3t}_{z(t)}\,\mathbf k](https://tex.z-dn.net/?f=C%3A%5Cmathbf%20r%28t%29%3D%5Cunderbrace%7B15%5Csin%20t%7D_%7Bx%28t%29%7D%5C%2C%5Cmathbf%20i%2B%5Cunderbrace%7B13%5Csin%5E2t%7D_%7By%28t%29%7D%5C%2C%5Cmathbf%20j%2B%5Cunderbrace%7B12%5Csin%5E3t%7D_%7Bz%28t%29%7D%5C%2C%5Cmathbf%20k)
with
![0\le t\le\dfrac\pi2](https://tex.z-dn.net/?f=0%5Cle%20t%5Cle%5Cdfrac%5Cpi2)
, and given the vector field
![\mathbf f(x,y,z)=x\,\mathbf i+y\,\mathbf j+z\,\mathbf k](https://tex.z-dn.net/?f=%5Cmathbf%20f%28x%2Cy%2Cz%29%3Dx%5C%2C%5Cmathbf%20i%2By%5C%2C%5Cmathbf%20j%2Bz%5C%2C%5Cmathbf%20k)
the work done by
![\mathbf f](https://tex.z-dn.net/?f=%5Cmathbf%20f)
on a particle moving on along
![C](https://tex.z-dn.net/?f=C)
is given by the line integral
![\displaystyle\int_C\mathbf f\cdot\mathrm d\mathbf r=\int\limits_{t=0}^{t=\pi/2}\mathbf f(x(t),y(t),z(t))\cdot\frac{\mathrm d\mathbf r(t)}{\mathrm dt}\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Cmathbf%20f%5Ccdot%5Cmathrm%20d%5Cmathbf%20r%3D%5Cint%5Climits_%7Bt%3D0%7D%5E%7Bt%3D%5Cpi%2F2%7D%5Cmathbf%20f%28x%28t%29%2Cy%28t%29%2Cz%28t%29%29%5Ccdot%5Cfrac%7B%5Cmathrm%20d%5Cmathbf%20r%28t%29%7D%7B%5Cmathrm%20dt%7D%5C%2C%5Cmathrm%20dt)
where
![\mathrm d\mathbf r=(15\cos t\,\mathbf i+26\sin t\cos t\,\mathbf j+36\sin^2t\cos t\,\mathbf k)\,\mathrm dt](https://tex.z-dn.net/?f=%5Cmathrm%20d%5Cmathbf%20r%3D%2815%5Ccos%20t%5C%2C%5Cmathbf%20i%2B26%5Csin%20t%5Ccos%20t%5C%2C%5Cmathbf%20j%2B36%5Csin%5E2t%5Ccos%20t%5C%2C%5Cmathbf%20k%29%5C%2C%5Cmathrm%20dt)
The integral is then
![\displaystyle\int_0^{\pi/2}(15\sin t\,\mathbf i+13\sin^2t\,\mathbf j+12\sin^3t\,\mathbf k)\cdot(15\cos t\,\mathbf i+13\sin2t\,\mathbf j+18\sin t\sin2t\,\mathbf k)\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_0%5E%7B%5Cpi%2F2%7D%2815%5Csin%20t%5C%2C%5Cmathbf%20i%2B13%5Csin%5E2t%5C%2C%5Cmathbf%20j%2B12%5Csin%5E3t%5C%2C%5Cmathbf%20k%29%5Ccdot%2815%5Ccos%20t%5C%2C%5Cmathbf%20i%2B13%5Csin2t%5C%2C%5Cmathbf%20j%2B18%5Csin%20t%5Csin2t%5C%2C%5Cmathbf%20k%29%5C%2C%5Cmathrm%20dt)
![=\displaystyle\int_0^{\pi/2}(432\sin^5t\cos t+338\sin^3t\cos t+225\sin t\cos t](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_0%5E%7B%5Cpi%2F2%7D%28432%5Csin%5E5t%5Ccos%20t%2B338%5Csin%5E3t%5Ccos%20t%2B225%5Csin%20t%5Ccos%20t)
Hey!
To solve x in this equation we must first add five to both sides to get
![\sqrt{x}](https://tex.z-dn.net/?f=%20%5Csqrt%7Bx%7D%20)
on its own.
<em>Original Equation :</em>
![\sqrt{x} - 5 = x - 11](https://tex.z-dn.net/?f=%20%5Csqrt%7Bx%7D%20-%205%20%3D%20x%20-%2011)
<em>New Equation {Added 5 to Both Sides} :</em>
![\sqrt{x} =x-6](https://tex.z-dn.net/?f=%20%5Csqrt%7Bx%7D%20%3Dx-6)
Now we must square both sides of the equation.
<em>Old Equation :</em>
![\sqrt{x} =x-6](https://tex.z-dn.net/?f=%20%5Csqrt%7Bx%7D%20%3Dx-6)
<em>New Equation {Changed by Squaring Both Sides} :</em>
![x= x^{2} -12x+36](https://tex.z-dn.net/?f=x%3D%20x%5E%7B2%7D%20-12x%2B36)
And now we must solve the new equation.
Step 1 - Switch sides
![x^{2} -12x+36=x](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20-12x%2B36%3Dx)
Step 2 - Subtract x from both sides
![x^{2} -12x+36-x=x-x](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20-12x%2B36-x%3Dx-x)
Step 3 - Simplify
![x^{2} -13x+36=0](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20-13x%2B36%3D0)
Now we need to solve the rest of the equation using the quadratic formula.
![\frac{-(-13)+ \sqrt{(-13) ^{2}-4*1*36 } }{2*1}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-%28-13%29%2B%20%5Csqrt%7B%28-13%29%20%5E%7B2%7D-4%2A1%2A36%20%7D%20%7D%7B2%2A1%7D%20)
![{13+ \sqrt{(-13)^{2}-4*1*36 }=13+ \sqrt{25}](https://tex.z-dn.net/?f=%7B13%2B%20%5Csqrt%7B%28-13%29%5E%7B2%7D-4%2A1%2A36%20%7D%3D13%2B%20%5Csqrt%7B25%7D)
![\frac{13+ \sqrt{25}}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B13%2B%20%5Csqrt%7B25%7D%7D%7B2%7D%20)
![\sqrt{25}=5](https://tex.z-dn.net/?f=%20%5Csqrt%7B25%7D%3D5%20)
![\frac{13+5}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B13%2B5%7D%7B2%7D%20)
![\frac{18}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B18%7D%7B2%7D%20)
9
![\frac{-(-13)- \sqrt{(-13) ^{2} -4*1*36} }{2*1}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-%28-13%29-%20%5Csqrt%7B%28-13%29%20%5E%7B2%7D%20-4%2A1%2A36%7D%20%7D%7B2%2A1%7D%20)
4
<em>So, this means that in the equation
![\sqrt{x} -5=x-11](https://tex.z-dn.net/?f=%20%5Csqrt%7Bx%7D%20-5%3Dx-11)
,</em>
x = 9 <em>and </em>
x = 4.Hope this helps!
- Lindsey Frazier ♥
(8q)/4 ..................
If you would like to solve 6 * (22 + 9), you can calculate this using the following steps:
6 * (22 + 9) = 6 * 31 = 186
The correct result would be 186.