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faltersainse [42]
3 years ago
11

Find the volume of the solid formed by revolving the region bounded by LaTeX: y = \sqrt{x} y = x and the lines LaTeX: y = 1 y =

1 and LaTeX: x = 4 x = 4 about the line LaTeX: y = 1 y = 1 .

Mathematics
1 answer:
Strike441 [17]3 years ago
6 0

Answer:

The volume is:

\displaystyle\frac{37\pi}{10}

Step-by-step explanation:

See the sketch of the region in the attached graph.

We set the integral using washer method:

\displaystyle\int_a^b\pi r^2dx

Notice here the radius of the washer is the difference of the given curves:

x-\sqrt{x}

So the integral becomes:

\displaystyle\int_1^4\pi(x-\sqrt{x})^2dx

We solve it:

Factor \pi out and distribute the exponent (you can use FOIL):

\displaystyle\pi\int_1^4x^2-2x\sqrt{x}+x\,dx

Notice: x\sqrt{x}=x\cdot x^{1/2}=x^{3/2}

So the integral becomes:

\displaystyle\pi\int_1^4x^2-2x^{3/2}+x\,dx

Then using the basic rule to evaluate the integral:

\displaystyle\pi\left[\frac{x^3}{3}-\frac{2x^{5/2}}{5/2}+\frac{x^2}{2}\right|_1^4

Simplifying a bit:

\displaystyle\pi\left[\frac{x^3}{3}-\frac{4x^{5/2}}{5}+\frac{x^2}{2}\right|_1^4

Then plugging the limits of the integral:

\displaystyle\pi\left[\frac{4^3}{3}-\frac{4(4)^{5/2}}{5}+\frac{4^2}{2}-\left(\frac{1}{3}-\frac{4}{5}+\frac{1}{2}\right)\right]

Taking the root (rational exponents):

\displaystyle\pi\left[\frac{4^3}{3}-\frac{4(2)^{5}}{5}+\frac{4^2}{2}-\left(\frac{1}{3}-\frac{4}{5}+\frac{1}{2}\right)\right]

Then doing those arithmetic computations we get:

\displaystyle\frac{37\pi}{10}

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3 years ago
What is the answer to Q=1/2p+15, for p​
Mazyrski [523]

Answer: Q=1/2p+15,= p=2q-30= Slope = 1.000/2.000 = 0.500

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Step-by-step explanation: Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                    p-(2*q-30)=0  

Solve   p-2q+30  = 0

we have an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes

x tells us how far along

m is the Slope or Gradient i.e. how steep the line is

b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line  p-2q+30  = 0 and calculate its properties

Notice that when p = 0 the value of q is 15/1 so this line "cuts" the q axis at q=15.00000

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When q = 0 the value of p is -30/1 Our line therefore "cuts" the p axis at p=-30.00000

 p-intercept = -30/1  = -30.00000

Slope is defined as the change in q divided by the change in p. We note that for p=0, the value of q is 15.000 and for p=2.000, the value of q is 16.000. So, for a change of 2.000 in p (The change in p is sometimes referred to as "RUN") we get a change of 16.000 - 15.000 = 1.000 in q. (The change in q is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

   Slope     =  1.000/2.000 =  0.500

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4÷25times4÷100times4÷4
notsponge [240]

Answer:

0.0064 is your answer

Step-by-step explanation:

because 4÷25 = 0.16 and 4÷100=0.04 and of course 4÷4=1

When the 0.16 and 0.04 that gets 0.0064

3 0
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