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