A few integration problems.. (1 Viewer)

Carrotsticks · Feb 23, 2012

Coookies said:
I need help finishing this question!

Also, a few volume Qs...
Q8. The parabola y=(x+2)^2 is rotated about the x-axis from x=0 to x=2. Find the volume of solid formed
Q19. Find the volume of solid formed when line x+3y-1=0 is rotated about x-axis from x=0 to x=8
Q20. I need help changing the subject. y=x^3, rotated about the y-axis. (So I need to get x^2)

Question 8:

$\pi \int_{0}^{2} (x+2)^{4} dx = \frac{\pi}{5} \left [ (x+2)^5 \right ]_0 ^2$

Question 19:

We must first make y the subject:

$\\ y=\frac{1}{3} (1-x) \\\\ V = \pi \int_{0}^{8} y^{2}dx = \pi \int_{0}^{8} {\frac{1}{9} (1-x) ^{2}}dx = \frac{\pi}{9} \int_{0}^{8} (1-x) ^{2}dx$

Question 20:

$\\ y=x^3 \\\\ $Cube root both sides$: \\\\ _{3}\sqrt{y} = x \Rightarrow x = _{3}\sqrt{y} \\\\ $Square both sides:$ \\\\ x^{2} = _{3}\sqrt{y^{2}} = y^{\frac{2}{3}}$

Excuse my typing, I don't know how to get nth root of something in LaTeX.

AAEldar · Feb 23, 2012

$\sqrt[n]{y^2}$

\sqrt[n]{y^2}

Carrotsticks · Feb 23, 2012

AAEldar said:
$\sqrt[n]{y^2}$

\sqrt[n]{y^2}

Many thanks AAEldar.

Drewk · Feb 27, 2013

Carrotsticks said:
$\\ y= \sqrt{4-x^{2}} \\\\ $This is the upper semi-circle from a circle with radius 2. However, we are finding the area of the region in the First Quadrant, so it will be a quarter of the area of the circle with radius 2.$ \\\\ $Area whole circle is:$ \\\\ A = \pi (2)^{2} = 4 \pi \\\\\ $However, we are finding the area of a quarter of it, so we divide our answer by 4:$ \\\\ A = \frac{1}{4} \times 4 \pi = \pi$

Hi i am stuck on the same question why cant i just use integration with limits 0 and 2 to get the right answer also getting 5.333

Carrotsticks · Feb 27, 2013

Drewk said:
Hi i am stuck on the same question why cant i just use integration with limits 0 and 2 to get the right answer also getting 5.333

You most certainly can use integration with limits from 0 to 2, but that would require a trigonometric substitution, which takes a while to evaluate usually (also opens situations for silly errors).

A few integration problems.. (1 Viewer)

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