Date | November 2018 | Marks available | 8 | Reference code | 18N.1.SL.TZ0.S_6 |
Level | Standard Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 0 |
Command term | Find | Question number | S_6 | Adapted from | N/A |
Question
Let f(x)=6−2x√16+6x−x2. The following diagram shows part of the graph of f.
The region R is enclosed by the graph of f, the x-axis, and the y-axis. Find the area of R.
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
METHOD 1 (limits in terms of x)
valid approach to find x-intercept (M1)
eg f(x)=0, 6−2x√16+6x−x2=0, 6−2x=0
x-intercept is 3 (A1)
valid approach using substitution or inspection (M1)
eg u=16+6x−x2, ∫306−2x√udx, du=6−2x, ∫1√u, 2u12,
u=√16+6x−x2, dudx=(6−2x)12(16+6x−x2)−12, ∫2du, 2u
∫f(x)dx=2√16+6x−x2 (A2)
substituting both of their limits into their integrated function and subtracting (M1)
eg 2√16+6(3)−32−2√16+6(0)2−02, 2√16+18−9−2√16
Note: Award M0 if they substitute into original or differentiated function. Do not accept only “– 0” as evidence of substituting lower limit.
correct working (A1)
eg 2√25−2√16, 10−8
area = 2 A1 N2
METHOD 2 (limits in terms of u)
valid approach to find x-intercept (M1)
eg f(x)=0, 6−2x√16+6x−x2=0, 6−2x=0
x-intercept is 3 (A1)
valid approach using substitution or inspection (M1)
eg u=16+6x−x2, ∫306−2x√udx, du=6−2x, ∫1√u,
u=√16+6x−x2, dudx=(6−2x)12(16+6x−x2)−12, ∫2du
correct integration (A2)
eg ∫1√udu=2u12, ∫2du=2u
both correct limits for u (A1)
eg u = 16 and u = 25, ∫25161√udu, [2u12]2516, u = 4 and u = 5, ∫542du, [2u]54
substituting both of their limits for u (do not accept 0 and 3) into their integrated function and subtracting (M1)
eg 2√25−2√16, 10−8
Note: Award M0 if they substitute into original or differentiated function, or if they have not attempted to find limits for u.
area = 2 A1 N2
[8 marks]