Question

Check my stuff, please.
\[\int_1^3 2\pi x((x^2-4x+7)-4(x-2)^2)dx\]

Answer 1

\[2\pi \int_1^3 (-3x^3+12x^2 -3x)dx\\=6\pi \int_1^3 (-x^3+4x^2-3x) dx\]
\[=6\pi*(\dfrac{-x^4}{4}+\dfrac{4x^3}{3}-\dfrac{3x^2}{2})~~from 1 ~~to~~3\]

Answer 2

plug the numbers in
\[6\pi *[(\dfrac{-81}{4}+\dfrac{108}{3}-\dfrac{27}{2})-(\dfrac{-1}{4}+\dfrac{4}{3}-\dfrac{3}{2})]\]

Answer 3

I am supposed to get 16pi but I got 48/3 pi

Answer 4

I did \(\dfrac{-81}{4}+\dfrac{1}{4} = \dfrac{-80}{4}=-20\)
\(\dfrac{108}{3}-\dfrac{4}{3}= \dfrac{104}{3}\)
\(dfrac{-27}{2}+\dfrac{3}{2}=\dfrac{-24}{2}=-12\)

Answer 5

Add them all I got a wrong answer :(

Answer 6

\[\int_1^32\pi x\big[(x^2-4x+7)-4(x-2)^2\big]\,\mathrm dx\\
=2\pi\int_1^3x\big[x^2-4x+7-4(x^2-4x+4)\big]\,\mathrm dx\\
=2\pi\int_1^3x\big[x^2-4x+7-4x^2+16x-16\big]\,\mathrm dx\\
=2\pi\int_1^3 x\big[-3x^2+12x-9\big]\,\mathrm dx\\
=2\pi\int_1^3 [-3x^3+12x^2-9x]\,\mathrm dx\\
=\dots\]

Answer 7

Yes, Sir

Answer 8

http://www.wolframalpha.com/input/?i=int+x%28%28x^2-4x%2B7%29-4%28x-2%29^2%29dx+x+from+1+to+3

Answer 9

Looks like Wolf ram is wrong

Answer 10

No, not that!! I let 2pi out , just put the integral to calculate, it gives me 8, then *2pi =16pi that is the answer from the back of the book

Answer 11

\[=2\pi\int_1^3 [-3x^3+12x^2-9x]\,\mathrm dx\\
=6\pi\int_1^3 [-x^3+4x^2-3x]\,\mathrm dx\\
=6\pi\left.\Big[-\frac14x^4+\frac43x^3-\frac32x^2\Big]\right|_1^3\\
=\dots\]

Answer 12

Yes

Answer 13

\[6\pi\Big[\left(\frac{-81}{4}+\frac{108}{3}-\frac{27}{2}\right)-\left(\frac{-1}{4}+\frac{4}{3}-\frac{3}{2}\right)\Big]\\
=6\pi\Big[\big(-20.5+36-13.5\big)-\left(-0.25+1\tfrac{1}{3}-1.5\right)\Big]\\
=6\pi\Big[\big(2.25\big)-\left(-0.75+\tfrac{1}{3}\right)\Big]\\
=6\pi\Big[2.25+0.75-\tfrac{1}{3}\Big]\\
=6\pi\Big[3-1/3\Big]\\
=\pi[18-2]\\=\dots
\]

Answer 14

oh man!!! Thank you so much. stupid me!! hihihihi

Answer 15

[the wolfram link forgot the 2π]

Answer 16

all the brackets negative signs and fractions make errors likely

Answer 17

I didn't put 2pi in, it's good.

Answer 18

Again, thank you "uncle" hihihihi

Answer 19

hiii!!!