What's is the verbal expression of 23f

Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant below the line y=5 and betw

vfiekz [6]

First, complete the square in the equation for the second circle to determine its center and radius:

x ² - 10x + y ² = 0

x ² - 10x + 25 + y ² = 25

(x - 5)² + y ² = 5²

So the second circle is centered at (5, 0) with radius 5, while the first circle is centered at the origin with radius √100 = 10.

Now convert each equation into polar coordinates, using

x = r cos(θ)

y = r sin(θ)

Then

x ² + y ² = 100 → r ² = 100 → r = 10

x ² - 10x + y ² = 0 → r ² - 10 r cos(θ) = 0 → r = 10 cos(θ)

y = 5 → r sin(θ) = 5 → r = 5 csc(θ)

See the attached graphic for a plot of the circles and line as well as the bounded region between them. The second circle is tangent to the larger one at the point (10, 0), and is also tangent to y = 5 at the point (0, 5).

Split up the region at 3 angles θ₁, θ₂, and θ₃, which denote the angles θ at which the curves intersect. They are

θ₁ = 0 … … … by solving 10 = 10 cos(θ)

θ₂ = π/6 … … by solving 10 = 5 csc(θ)

θ₃ = 5π/6 … the second solution to 10 = 5 csc(θ)

Then the area of the region is given by a sum of integrals:

$\displaystyle \frac12\left(\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\}\left(10^2-(10\cos(\theta))^2\right)\,\mathrm d\theta+\int_{\frac\pi6}^{\frac{5\pi}6}\left((5\csc(\theta))^2-(10\cos(\theta))^2\right)\,\mathrm d\theta\right)$

$=\displaystyle 50\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\} \sin^2(\theta)\,\mathrm d\theta+\frac12\int_{\frac\pi6}^{\frac{5\pi}6}\left(25\csc^2(\theta) - 100\cos^2(\theta)\right)\,\mathrm d\theta$

To compute the integrals, use the following identities:

sin²(θ) = (1 - cos(2θ)) / 2

cos²(θ) = (1 + cos(2θ)) / 2

and recall that

d(cot(θ))/dθ = -csc²(θ)

You should end up with an area of

$=\displaystyle25\left(\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\}(1-\cos(2\theta))\,\mathrm d\theta-\int_{\frac\pi6}^{\frac{5\pi}6}(1+\cos(2\theta))\,\mathrm d\theta\right)+\frac{25}2\int_{\frac\pi6}^{\frac{5\pi}6}\csc^2(\theta)\,\mathrm d\theta$

$=\boxed{25\sqrt3+\dfrac{125\pi}3}$

We can verify this geometrically:

• the area of the larger circle is 100π

• the area of the smaller circle is 25π

• the area of the circular segment, i.e. the part of the larger circle that is bounded below by the line y = 5, has area 100π/3 - 25√3

Hence the area of the region of interest is

100π - 25π - (100π/3 - 25√3) = 125π/3 + 25√3

as expected.

3 0

3 years ago

Can someone please help me?

bulgar [2K]

Solution:

Given:

18 = 2b

Solve the equation by isolating the variable.

=> 18 = 2b
=> 2b/2 = 18/2
=> b = 9

The value of b is 9.

Hoped this helped!

6 0

2 years ago

8+y−23k+3m What is the constant? What is the coefficient of the y term? How many terms are there?

Dimas [21]

Answer:

Step-by-step explanation:

There are 4 terms

Coefficient of y term = 1

Constant = 8

The term without variable is the constant term.

4 0

3 years ago

A car is speeding at 60 miles per hour along a winding coastal highway. (The posted speed limit is 45 miles perhour). The speedi

Dmitry [639]

8 minutes 20 seconds. First, lets determine who many miles per minute each vehicle moves by dividing each speed by 60. Speeder = 60 / 60 = 1 mile per minute. Police = 75 / 60 = 1.25 miles per minute. Other car = 45 / 60 = 0.75 miles per minute. Since the speeding car moves for 5 minutes before the police start to chase, that means that the speeding car will now be 5 + T miles down the road with T being the time the police has been chasing. The police will be 1.25 T. We're looking for when those two equations equal each other. So 5 + T = 1.25 T Subtract T from both sides 5 = 0.25T Divide both sides by 0.25 20 = T So it will take the police officer 20 minutes to catch up to the speeder. And they will both have traveled a total distance of 25 miles from the point where the speeder passed the police car. Now we need to figure out how far the law obeying car has moved during those 25 minutes. So 25 * 0.75 = 18.75 miles. The distance the law obeying car needs to travel to catch up to the police officer then becomes 25 - 18.75 = 6.25 miles. The number of minutes that the law obeying car needs to travel that distance is 6.25 / 0.75 = 8.333.... Which is 8 minutes 20 seconds.

8 0

4 years ago

Holly had 5,000 in her bank account she withdrew 800 to buy a new bike what is the percent decrease in the balance of her accoun

Anika [276]

800/5000*100=16%
Holly's account decreases on 16 percents

6 0

3 years ago