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Both, for different reasons!.

estimate the area under the curve y(x) = ln(x) from x=1 to x=2

\(\boxed{~ \int \limits_{x=1}^{x=2} \ln{(x)}\ dx = \ ? ~ }\)

\(\text{Integration by parts } \boxed{~ \int u'v= uv - \int uv' ~} \\ \begin{array}{rclcrcl} u' &=& 1 && v&=& \ln{(x)} \\ u &=& x && v' &=& \frac{1}{x}\\\\ \int 1\cdot \ln{(x)} \ dx &=& x\cdot \ln{(x)} -\int x\cdot \frac{1}{x} \ dx\\ &=& x\cdot \ln{(x)} -\int \ dx\\ \int 1\cdot \ln{(x)} \ dx &=& x\cdot \ln{(x)} - x\\\\ \end{array}\\ \begin{array}{rcl} \int \limits_{x=1}^{x=2} \ln{(x)}\ dx &=& \left[ x\cdot \ln{(x)} - x \right]_{1}^{2}=2\cdot \ln{(2)-2} - [ 1\cdot \ln{(1)}-1] \qquad | \qquad \ln{(1)} = 0 \\ &=& 2\cdot \ln{(2)}-2 - (-1)\\ &=& 2\cdot \ln{(2)}-2 +1\\ &=& 2\cdot \ln{(2)}-1\\ \mathbf{ \int \limits_{x=1}^{x=2} \ln{(x)}\ dx }& \mathbf{=} & \mathbf{ 0.38629436112 } \end{array}\)

The area under the curve y(x) = ln(x) from x=1 to x=2 is 0.38629436112

Have a look at the graph here.........https://www.desmos.com/calculator/wk4fnvhslz

The  area bounded by x= 1, x = 2 and the x axis   is almost a triangle  with a base of 1 and a height of  about 7/ 10

So the area [ estimated] is about (1/2)(1)(7/10)  = 7/20  = 0.35 sq units

BTW ....The actual value is about 0.38629 sq units

A space shuttle 200 miles above the earth is orbiting the earth once every 12 hours. How far does the shuttle travel in 1 hour? (Assume the radius of the earth is 4,000 miles.) Give your answer as both an exact value and an approximation to three significant digits

orbit = \(2\pi \cdot (4000+200)\ \text{miles} = 2\pi\cdot 4200 \ \text{miles}\)

velocity = \(\frac{orbit}{12 \ \text{hours} } = \frac{ 2\pi \cdot 4200\ \text{miles} } {12 \ \text{hours} } = \frac{ \pi \cdot 4200\ \text{miles} } { 6 \ \text{hours} } = 700\pi \cdot \frac{ \text{miles} } { \text{hour} } \)

The shuttle travel in 1 hour

\(700\cdot\pi \ \text{miles} = 2199.11485751\ \text{miles} \approx2199.115 \ \text{miles} \)

y=3sin(6x−π)   we can write this as.....

y = 3sin[6(x - pi/6)]      there are 6 periods in 2pi rads and the phase shift is pi/6 rads to the right

6(-2)-14/(-7)  =

-12 - [-2]  =

-12 + 2 =

-10

Edit ....[ Sorry..... I hit the wrong buttons on the calculator before   !!!]

The distance traveled in one hour is given by:

d= r[theta]  =  4200 [ 2pi/12]   = 4200[pi / 6]  = about 2199.1148575128552669  mi/hr  =

2199.115 mi/hr   to 3 sig. decimal digits

53/4-47/8

=(106-47)/8

=59/8

To prove this add  59/8 to 47/8  and divide the numerator and denominator of the result by 2

thanks

It's actually a repeating decimal with a length of 46 digits......this is the max length of repeating digits for a fraction with a denominator of 47

The answer is :

         3.714285714285........

80 92 91 75 89 84 75 and 85   so we have

(80 + 92 + 91 + 75 + 89 + 84 + 75 + 85)  / 8 =   [671 ] /  [8 ] = about 83.9

P( 6 , -5); Q(-3,  8)

  ( x1, y1)    (x2, y2)

The slope is given by

[ y2 - y1 ] / [ x2 - x1]   =   [8 - (-5)] / [ -3 - 6]  =  13 / [-9]   =   -13 / 9

∫ 1/ sqrt(2x) dx =

∫ (2x)^-1/2  dx         let 2x = u   →  du= 2dx  →  du/2 = dx   so we have

(1/2) ∫ u^-1/2 du   =

(1/2) *(2) * u^1/2   +  C  =

(2x)^1/2 + C

Sec (2a+10°)=csc (3a-40°)

These are cofunctions.....this means that.....

2a + 10 + 3a - 40   =  90    simplify

5a -30  = 90    add 30 to both sides

5a  = 120    divide both sides by 5

24  = a      

The greatest number is twice as large as the least number, and the middle number is three more than the least number, and the sum is 75?

Let the the least number = x

The greatest number = 2x

The middle number = x + 3

And we have

x + [x + 3 ] + 2x = 75   simplify

4x + 3 =   75     subtract 3 from both sides

4x = 72    divide both sides by 4

x = 18   this is the least

2x = 2(18)  = 36    this is the greatest

x + 3  =  18 + 3 = 21    this is the middle

Note 

sin^-1 (sin[theta])   = theta   

So

sin-1(sin6pi/7)  =   [ 6pi ] /7

Answer: -5/6

4

1/2^30/17.5

Is this...

a) (1/2)^(30/17.5)

b) [(1/2)^30]/17.5

c) [1/(2^30)]/17.5

d) 1/[2^(30/17.5)]

The calculator will solve this using order of operations so if there are no directional markings, such as parenthesis, then the calculator will probably not give you the answer you are looking for.

7 + 2/3(x) = -7

Since you can easily see that your end result is the opposite of one of the two parts on the opposite side of the equal sign, you at least know what your, in this case, middle number should be. Meaning, in order to get from 7 to -7 you know the number between them is 14 and that the 14 has to be negative because the number is being added to the +7 to get to -7.

What I would do, knowing this information, is multiply 14 by 3 (= 42) and then divide by 2 (21). Since you know it has to be negative you can just make the 21 negative. You can then substitute x for -21 and see that you get an answer of -7.

7 + 2/3(-21) = -7

7 + (2*-21)/3 = -7

7 + (-42)/3 = -7

7 - 14 = -7

-7 = -7

8*3/4

 Basically 8/1 * 3/4  => 8*3/4 = 24/4 = 6

Any whole number can be put over 1 and, for this case, multiplied across to be simplified and then you have your answer.

y=14x^2z

Piece 1) 5 3/4 feet => 5 6/8 feet

Piece 2) 4 7/8 feet

5 6/8 feet - 4 7/8 feet = 7/8

To prove this you can add 1 to 4 7/8 and see that you'll have 5 7/8 and then subtract 1/8 from 1 and be left with the difference between the two lengths as 7/8 feet.

9+10= 19