[SPM Add Maths Integration] Chapter 3: integration area

Someone please help me :pleading_face::pray::pray::pray::pray::pleading_face:
Thank you so much

Question

  1. Rajah menunjukkan garis lurus A C menyilang lengkung y=x^{2}-6 x+15 pada titik A dan titik B.
    Diagram shows the straight line AC intersects the curve y=x^{2}-6 x+15 at point A and point B.
    image

Rajah / Diagram
(a) Hitung luas rantau berlorek P.
Calculate the area of the shaded region P.
[5 markah/marks]
Jawapan / Answer: 45 \frac{7}{12} \mathrm{~cm}^{2}
(b) Diberi bahawa \int_{0}^{4}\left(x^{2}-6 x+15\right) d x=p. Cari isi padu janaan, dalam sebutan p, apabila rantau berlorek Q diputarkan melalui 360^{\circ} pada paksi- x.
It is given that \int_{0}^{4}\left(x^{2}-6 x+15\right) d x=p. Find the volume generated, in terms of p. when the shaded region Q is rotated through 360^{\circ} about the x-axis.
[5 markah/marks]
Jawanan/Answer: \left(505 \frac{1}{3}-p\right) \pi unit^3

Solution

Hello! Whenever you come across integration questions, ask yourself: how do I break down the shapes into several parts in a way that is easy for me to solve?
This is how I split the shapes:
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The triangle seems easy to solve right? We just need the coordinate of C.
For the area shaded in blue, it’s also fairly simple: integrate the curve with respect to x from 0 to 4.

But we don’t know the coordinate of C, so we’ll start with the blue area first.
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All we have left is the red triangle, but firstly, how to find the coordinate of C? Well, A, B and C lies on the same line and notice that at the beginning of the question, it is mentioned that: curve intersects at point A. So we can find the coordinate of A from there. A lies on the y-axis, so x-coordinate of A is 0, substitute x=0 into the equation of the curve and solve for the y-coordinate. (This value of y-coordinate is also the y-intercept of line ABC.) From there, we can then find coordinate C using the equation of a straight line concept.
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The last two steps are pretty simple: find the area of the triangle and add the two areas!
image

Hope this helps! Will upload the solution to part b in a while! :slight_smile:

1 Like

Hello again Dhaksheina, sorry for the rather late reply,
I do think there is a high chance that the question is missing a square outside the bracket, but it is best if you could confirm it with your teacher!
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Here is my solution to part b:
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I personally prefer using the volume of a cone formula as it is quicker and harder to make errors, but you could also integrate the straight line to find the volume! :slight_smile:

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Hope this helps! :slight_smile: