Answer:
The answer is b
Step-by-step explanation:
1 quart is equal to 2 pints, so 3/4ths qt is equal to 1.5 pints
Answer:
∫ F dr= 1
c
Step-by-step explanation:
The correct question is -
Given -
and C : (0, 1) to ![(\frac{\pi }{4} , 1)](https://tex.z-dn.net/?f=%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%2C%201%29)
To find - Evaluate and input the result of the following integral.
Proof -
We know that,
dr = dx i + dy j
F = < F1, F2 >
Now, we know
![d(\frac{tanx}{y }) = \frac{d}{dx}(\frac{tanx}{y}) + \frac{d}{dy}(\frac{tanx}{y})](https://tex.z-dn.net/?f=d%28%5Cfrac%7Btanx%7D%7By%20%7D%29%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%28%5Cfrac%7Btanx%7D%7By%7D%29%20%20%2B%20%5Cfrac%7Bd%7D%7Bdy%7D%28%5Cfrac%7Btanx%7D%7By%7D%29)
So,
![\frac{d}{dx}(\frac{tanx}{y}) =\frac{1}{y} \frac{d}{dx}({tanx}) \\\\ = \frac{sec^{2} x}{y} \\\frac{d}{dy}(\frac{tanx}{y}) = tanx \frac{d}{dy}(y^{-1}) \\ = -\frac{tanx}{y^{2} }](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%28%5Cfrac%7Btanx%7D%7By%7D%29%20%3D%5Cfrac%7B1%7D%7By%7D%20%5Cfrac%7Bd%7D%7Bdx%7D%28%7Btanx%7D%29%20%20%5C%5C%5C%5C%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3D%20%5Cfrac%7Bsec%5E%7B2%7D%20x%7D%7By%7D%20%5C%5C%5Cfrac%7Bd%7D%7Bdy%7D%28%5Cfrac%7Btanx%7D%7By%7D%29%20%3D%20tanx%20%5Cfrac%7Bd%7D%7Bdy%7D%28y%5E%7B-1%7D%29%20%20%5C%5C%20%20%20%20%3D%20-%5Cfrac%7Btanx%7D%7By%5E%7B2%7D%20%7D)
Now,
We can see that,
![d(\frac{tanx}{y }) = \frac{sec^{2} x}{y} dx - (\frac{tanx}{y^{2} })dy](https://tex.z-dn.net/?f=d%28%5Cfrac%7Btanx%7D%7By%20%7D%29%20%3D%20%5Cfrac%7Bsec%5E%7B2%7D%20x%7D%7By%7D%20dx%20-%20%28%5Cfrac%7Btanx%7D%7By%5E%7B2%7D%20%7D%29dy)
So,
![\int\limits^{}_C {F. dr} = \int\limits^{}_C d(\frac{tanx}{y} )\\= [\frac{tanx}{y} ]\limits^{(\frac{\pi }{4} ,1)}_{(0,1)}\\= [\frac{tanx}{y} ]\limits^{}_{(\frac{\pi }{4} ,1)} - [\frac{tanx}{y} ]\limits^{}_{(0,1)}\\= \frac{tan(\frac{\pi }{4} )}{1} - \frac{tan 0}{1}\\= 1 - 0\\= 1](https://tex.z-dn.net/?f=%5Cint%5Climits%5E%7B%7D_C%20%7BF.%20dr%7D%20%3D%20%20%5Cint%5Climits%5E%7B%7D_C%20d%28%5Cfrac%7Btanx%7D%7By%7D%20%29%5C%5C%3D%20%5B%5Cfrac%7Btanx%7D%7By%7D%20%5D%5Climits%5E%7B%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%2C1%29%7D_%7B%280%2C1%29%7D%5C%5C%3D%20%5B%5Cfrac%7Btanx%7D%7By%7D%20%5D%5Climits%5E%7B%7D_%7B%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%2C1%29%7D%20-%20%5B%5Cfrac%7Btanx%7D%7By%7D%20%5D%5Climits%5E%7B%7D_%7B%280%2C1%29%7D%5C%5C%3D%20%5Cfrac%7Btan%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%29%7D%7B1%7D%20-%20%5Cfrac%7Btan%200%7D%7B1%7D%5C%5C%3D%201%20-%200%5C%5C%3D%201)
∴ we get
∫ F dr= 1
c
It’s -1/3 because you subtract 1-5 and -3-9, you get -4/-12 then simplify to get -1/3.