Anyone knows how to do this? Thanks

## Question

- Diagram below shows the straight line P Q and the straight line y=2 intersecting the curve y^{2}=x-2 at point Q. /
Rajah di bawah menunjukkan garis lurus P Q dan garis lurus y=2 bersilang lengkung y^{2}=x-2 pada titik Q.

It is given that the area of the shaded region is 12 \frac{2}{3} unit ^{2}.

Diberi bahawa luas rantau berlorek ialah 12 \frac{2}{3} unit { }^{2}.

a) Find the value of k. /Cari nilai k.

b) The region enclosed by the curve, the straight line y=2, the x-axis and the y axis, is revolved through 360^{\circ} about the y axis. Find the volume of revolution in terms of \pi.

### Answer

Hi weiling,

For part (a), we need to split the area of shaded region into two parts:

for the blue triangle’s area, we would use the following formula: \frac{1}{2} \times length \times height

height is (k-2), for length we just need to find the x-coordinate where the curve y^2=x-2 and line y=2 intersect.

Substituting y=2 into the equation of the curve gives:

2^2=x-2

4=x-2

x=6

And so, the area of the triangle is \frac{1}{2} \times 6 \times (k-2) = 3(k-2)

For the area shaded in red, we would need to **integrate the curve with respect to y**.

The general formula is \int_a^b x \space dy

Substituting a (lower bound) as 0 and b (upper bound) as 2, we get:

blue area + red area = total area

Again, we use substitution to solve for k.

## (b)

For part (b), **the volume is not the same region as in part (a) revolved around y-axis**. If you read the question carefully, it only wants the **red area in part (a) revolved around y-axis.**

The general formula for volume (revolved around y-axis) is \pi \int_a^b x^2 \space dy

Again, we substitute a (lower bound) as 0 and b (upper bound) as 2:

*****edit**: should be unit ^3 not unit ^2

Hope this helps! Feel free to clarify if you don’t understand anything!

Thank you so much for ur clear explaination

Is the keyword here about the y axis (finding area revolution) but what does enclosed mean about the curve

Hi maryah, apologies for the late reply, I’m not too sure what you mean but “enclosed” basically mean “surrounded”

So for b), the region is actually enclosed/surrounded on 4 sides: (with each side representing a different colour as drawn in the diagram)

- curve (blue)
- straight line y=2 (green)
- x-axis (red)
- y-axis (yellow)

I think the keyword would be finding the volume of revolution, but first you need to know which region the question is asking for and whether if it’s revolution about x/y-axis. Hope this helps