Solve for x. Round your answer to the nearest tenth if necessary.

find the volume of the solid formed by revolving the region bounded by the graphs of y = 4x - x^2 and f(x) = x^2 from [0,2] abou

Neko [114]

Answer:

$v = \frac{32\pi}{3}$

or

$v=33.52$

Step-by-step explanation:

Given

$f(x) = 4x - x^2$

$g(x) = x^2$

$[a,b] = [0,2]$

Required

The volume of the solid formed

Rotating about the x-axis.

Using the washer method to calculate the volume, we have:

$\int dv = \int\limit^b_a \pi(f(x)^2 - g(x)^2) dx$

Integrate

$v = \int\limit^b_a \pi(f(x)^2 - g(x)^2)\ dx$

$v = \pi \int\limit^b_a (f(x)^2 - g(x)^2)\ dx$

Substitute values for a, b, f(x) and g(x)

$v = \pi \int\limit^2_0 ((4x - x^2)^2 - (x^2)^2)\ dx$

Evaluate the exponents

$v = \pi \int\limit^2_0 (16x^2 - 4x^3 - 4x^3 + x^4 - x^4)\ dx$

Simplify like terms

$v = \pi \int\limit^2_0 (16x^2 - 8x^3 )\ dx$

Factor out 8

$v = 8\pi \int\limit^2_0 (2x^2 - x^3 )\ dx$

Integrate

$v = 8\pi [ \frac{2x^{2+1}}{2+1} - \frac{x^{3+1}}{3+1} ]|\limit^2_0$

$v = 8\pi [ \frac{2x^{3}}{3} - \frac{x^{4}}{4} ]|\limit^2_0$

Substitute 2 and 0 for x, respectively

$v = 8\pi ([ \frac{2*2^{3}}{3} - \frac{2^{4}}{4} ] - [ \frac{2*0^{3}}{3} - \frac{0^{4}}{4} ])$

$v = 8\pi ([ \frac{2*2^{3}}{3} - \frac{2^{4}}{4} ] - [ 0 - 0])$

$v = 8\pi [ \frac{2*2^{3}}{3} - \frac{2^{4}}{4} ]$

$v = 8\pi [ \frac{16}{3} - \frac{16}{4} ]$

Take LCM

$v = 8\pi [ \frac{16*4- 16 * 3}{12}]$

$v = 8\pi [ \frac{64- 48}{12}]$

$v = 8\pi * \frac{16}{12}$

Simplify

$v = 8\pi * \frac{4}{3}$

$v = \frac{32\pi}{3}$

or

$v=\frac{32}{3} * \frac{22}{7}$

$v=\frac{32*22}{3*7}$

$v=\frac{704}{21}$

$v=33.52$

8 0

2 years ago

Ryan went to the store to buy a CD. The CD was on sale for 10% off. If Ryan paid $13.50 for the CD, what was the original price

nydimaria [60]

<span>well first off, you take 1350/9=150
then you add it all together 1350+150=1500
then your full answer would be $15 was the original price
</span>

3 0

3 years ago

Word problem:

shepuryov [24]

The race was 2.5 miles because an hour divided by 2 is 30 minutes. Therefore you have to do the same thing to the 5 and 2.5 multiplied by 2 is 5.

7 0

3 years ago

you have 28 dog biscuit. You have 8 dog bowls. Each dog bow gets the same number of biscuits. how many biscuit are left over?

bearhunter [10]

First you want to find a multiple that's very close to 28. It's 24, so subtract 28-24 which is 4. 4 biscuits are left.

Hope that helped:)

5 0

2 years ago

Read 2 more answers

Indicate in standard form the equation of the line passing through the given point and having the given slope.. . B(6, 2), m = -

Leni [432]

General form of Equation of straight line is:
y=m x + b
m=-1/2 passing through (6,2)
plug these values in general form,we get
2=-1/2(6)+b
2=-3+b
5=b
Hence,
y=(-1/2)x + 5
Rewriting to make it in General form:
-1/2 x + y - 5 = 0

4 0

3 years ago