Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js

User interface language: English | Español

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)=62x16+6xx2. 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,  62x16+6xx2=0,  62x=0

x-intercept is 3      (A1)

valid approach using substitution or inspection      (M1)

eg   u=16+6xx2,  3062xudx,  du=62x1u,  2u12,

u=16+6xx2,   dudx=(62x)12(16+6xx2)12,  2du,  2u

f(x)dx=216+6xx2      (A2)

substituting both of their limits into their integrated function and subtracting      (M1)

eg   216+6(3)32216+6(0)202,  216+189216

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   225216,  108

area = 2      A1 N2

 

 

METHOD 2 (limits in terms of u)

valid approach to find x-intercept      (M1)

eg   f(x)=0,  62x16+6xx2=0,  62x=0

x-intercept is 3      (A1)

valid approach using substitution or inspection      (M1)

eg   u=16+6xx2,  3062xudx,  du=62x1u

u=16+6xx2,   dudx=(62x)12(16+6xx2)12,  2du

correct integration      (A2)

eg   1udu=2u12,   2du=2u

both correct limits for u      (A1)

eg   u = 16 and u = 25,  25161udu,   [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   225216,  108

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]

Examiners report

[N/A]

Syllabus sections

Topic 5 —Calculus » SL 5.10—Indefinite integration, reverse chain, by substitution
Show 54 related questions
Topic 5 —Calculus

View options