1. ## Integration

Sorry for reposting...I think my previous post could not be viewed. X.x

Q: A curve C is defined by the following pair of parametric equations:

$\displaystyle x = 1 + t^2$ and $\displaystyle y = \displaystyle{\frac{3 - t}{2+ t}}$ for 2 \1oq t \1oq 4. The region R is bounded by C, the x-axis and the lines x = 5 and x = 17.

a) Find the exact area of the region R.

b) Find V, the volume of solid generated when R is rotated through $\displaystyle 2\pi$ about the x-axis.

2. Originally Posted by Tangera
Sorry for reposting...I think my previous post could not be viewed. X.x

Q: A curve C is defined by the following pair of parametric equations:

$\displaystyle x = 1 + t^2$ and $\displaystyle y = \displaystyle{\frac{3 - t}{2+ t}}$ for 2 \1oq t \1oq 4. The region R is bounded by C, the x-axis and the lines x = 5 and x = 17.

a) Find the exact area of the region R.

b) Find V, the volume of solid generated when R is rotated through $\displaystyle 2\pi$ about the x-axis.

a) Area = $\displaystyle \int_{t_1}^{t_2} y \, dx$.

b) $\displaystyle V = \pi \int_{t_1}^{t_2} y^2 \, dx$.

In each case you substitute $\displaystyle y = \frac{3-t}{2+t}$ and $\displaystyle x = 1 + t^2 \Rightarrow dx = 2t \, dt$.

And since you're bounded by the x-axis it follows that $\displaystyle x = 5 \Rightarrow t_{1} = 2$ and $\displaystyle x = 17 \Rightarrow t = 4$.

3. Originally Posted by Tangera
Sorry for reposting...I think my previous post could not be viewed. X.x

Q: A curve C is defined by the following pair of parametric equations:

$\displaystyle x = 1 + t^2$ and $\displaystyle y = \displaystyle{\frac{3 - t}{2+ t}}$ for 2 \1oq t \1oq 4. The region R is bounded by C, the x-axis and the lines x = 5 and x = 17.

a) Find the exact area of the region R.

b) Find V, the volume of solid generated when R is rotated through $\displaystyle 2\pi$ about the x-axis.