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3 years ago

Help me with this problem?

Mathematics

1 answer:

nignag [31]3 years ago

4 0

Answer: $\frac{2x(5x+1)}{4x-1}$

Step-by-step explanation:

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Consider the line integral Z C (sin x dx + cos y dy), where C consists of the top part of the circle x 2 + y 2 = 1 from (1, 0) t

pashok25 [27]

Direct computation:

Parameterize the top part of the circle $x^2+y^2=1$ by

$\vec r(t)=(x(t),y(t))=(\cos t,\sin t)$

with $0\le t\le\pi$ , and the line segment by

$\vec s(t)=(1-t)(-1,0)+t(2,-\pi)=(3t-1,-\pi t)$

with $0\le t\le1$ . Then

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)$

$=\displaystyle\int_0^\pi(-\sin t\sin(\cos t)+\cos t\cos(\sin t)\,\mathrm dt+\int_0^1(3\sin(3t-1)-\pi\cos(-\pi t))\,\mathrm dt$

$=0+(\cos1-\cos2)=\boxed{\cos1-\cos2}$

Using the fundamental theorem of calculus:

The integral can be written as

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=\int_C\underbrace{(\sin x,\cos y)}_{\vec F}\cdot\underbrace{(\mathrm dx,\mathrm dy)}_{\vec r}$

If there happens to be a scalar function $f$ such that $\vec F=\nabla f$ , then $\vec F$ is conservative and the integral is path-independent, so we only need to worry about the value of $f$ at the path's endpoints.

This requires

$\dfrac{\partial f}{\partial x}=\sin x\implies f(x,y)=-\cos x+g(y)$

$\dfrac{\partial f}{\partial y}=\cos y=\dfrac{\mathrm dg}{\mathrm dy}\implies g(y)=\sin y+C$

So we have

$f(x,y)=-\cos x+\sin y+C$

which means $\vec F$ is indeed conservative. By the fundamental theorem, we have

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=f(2,-\pi)-f(1,0)=-\cos2-(-\cos1)=\boxed{\cos1-\cos2}$

3 0

3 years ago

Solve for x in the equation x2+2x+1=17

Sphinxa [80]

If you meant 2x + 2x + 1 = 17: x = 4
Solve for x by simplifying both sides of the equation, then isolating the variable.

If you meant x2 + 2x + 1 = 17: x ≈ 3.1231056,−5.1231056x
Solve the equation for x by finding a, b, and c of the quadratic then applying the quadratic formula. x = −1 ± √17x

7 0

4 years ago

Read 2 more answers

The perimeter of a rectangle is 300 feet. The width of the rectangle is 10 feet more than the length. What is the width of the r

gavmur [86]

300 = 2(x + x + 10)
300 = 2(2x +10)
300 = 4x + 20
4x = 280
x = 70
x + 10 = 80
width = 80ft
answer D) 80 feet

7 0

3 years ago

What is 8 lb. 12 oz. − 5 lb. 5 oz. ?

marysya [2.9K]

3 7 is the answer I hope this helps

7 0