Free Areas questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Calculus. Each question has the answer and the marking points an examiner would look for.
Generate a full Areas worksheet →All Calculus questionsThe region enclosed by the graph of y = e0.5x, the x-axis, the y-axis and the line x = k, where k > 0, has area 8.
Find the value of k.
| Part | Answer and marking points | Marks |
|---|---|---|
| k = 3.22 ∫0ke0.5xdx = 8 M1A1 attempt to solve using GDC (M1) k = 3.22 A1 | 4 |
The curve y = x2 − 5x − 2 and the line y = −3x − 2 intersect at two points.
Find the x-coordinates of the points of intersection.
Find the exact area of the region enclosed by the curve and the line.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | x = 0, x = 2 x2 − 2x = 0 (M1) x = 0, x = 2 A1 | 2 |
| (b) | 43 ∫02((−3x − 2) − (x2 − 5x − 2))dx M1 ∫02(−x2 + 2x)dx A1 correct integration and substitution (A1) 43 A1 | 4 |
Let f(x) = 5 − x2 and g(x) = 2x. The graphs of f and g intersect at x = p and x = q, where p < q are the two smallest solutions of f(x) = g(x).
Find the value of p and the value of q.
Find the area of the region enclosed by the graphs of f and g between x = p and x = q.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | p = −2.19, q = 1.48 p = −2.19 A1 q = 1.48 A1 | 2 |
| (b) | 10.1 integral of the difference of the functions (M1) correct limits (A1) 10.1 A1 | 3 |
The region enclosed by the graph of y = e0.5x, the x-axis, the y-axis and the line x = k, where k > 0, has area 5.
Find the value of k.
| Part | Answer and marking points | Marks |
|---|---|---|
| k = 2.51 ∫0ke0.5xdx = 5 M1A1 attempt to solve using GDC (M1) k = 2.51 A1 | 4 |
The curve y = x2 + x − 4 and the line y = x intersect at two points.
Find the x-coordinates of the points of intersection.
Find the exact area of the region enclosed by the curve and the line.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | x = −2, x = 2 x2 − 4 = 0 (M1) x = −2, x = 2 A1 | 2 |
| (b) | 323 ∫-22((x) − (x2 + x − 4))dx M1 ∫-22(−x2 + 4)dx A1 correct integration and substitution (A1) 323 A1 | 4 |
Let f(x) = 4cos x and g(x) = x2 − 2. The graphs of f and g intersect at x = p and x = q, where p < q are the two smallest solutions of f(x) = g(x).
Find the value of p and the value of q.
Find the area of the region enclosed by the graphs of f and g between x = p and x = q.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | p = −1.50, q = 1.50 p = −1.50 A1 q = 1.50 A1 | 2 |
| (b) | 11.7 integral of the difference of the functions (M1) correct limits (A1) 11.7 A1 | 3 |
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Meet Riyaz →Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in