IB Maths AA SLTopic 5 — CalculusPaper 2 (calc only)~7 min read
Finding Areas Using a GDC
A definite integral calculates the exact area between a curve and the x-axis. With a GDC, this is one of the easiest things in calculus — set the function and limits, hit enter. Just be careful: GDCs love giving decimals when an exact answer is required.
📘 What you need to know
Definite integral = area between the curve and x-axis from x = a to x = b.
Notation: A = ∫ab f(x) dx.
Limits come from vertical lines x = a, x = b, OR from where the curve crosses the x-axis (solve f(x) = 0).
No “+ c“ needed — it cancels in F(b) − F(a).
GDC gives decimals by default — convert back to exact form if the question asks.
What is the area under a curve?
The shaded region between a curve, the x-axis, and two vertical lines
The area A is bounded by four things: the curve y = f(x) on top, the x-axis on the bottom, and two vertical lines x = a and x = b on the sides.
The definite integral
Definite integral (area)
A = ∫ab f(x) dx = F(b) − F(a)
✓ in formula booklet (Fundamental Theorem of Calculus)
“Definite” means the answer is a number, not a function. The “+ c” cancels because it appears in both F(b) and F(a) and they subtract.
Setting up the integral
3-step setup
Sketch the curve and the area you want — use your GDC’s graphing screen.
Identify the limits a and b. They come either from given vertical lines, or from where f(x) = 0.
Write the integral: A = ∫(a to b) f(x) dx, then evaluate on your GDC.
📍
Limits from f(x) = 0
If the area is bounded by the curve crossing the x-axis (no vertical lines given), find the roots of f(x) = 0 — these are your limits. Use your GDC’s solver or graphing screen.
How to use your GDC
Calculator workflow
1
Find the integral function. Look for the ∫□□□ button — physical or in a menu.
2
Enter the integrand f(x) — the function to integrate.
3
Enter the lower limita and upper limitb.
4
Press enter. Read the answer — usually a decimal.
5
Convert to exact form if the question asks (decimal → fraction or surd).
🧠
“GDC = decimal · Exam = exact”
If the question says “exact answer” or “in the form pq“, a decimal won’t get the marks. Convert to a fraction or simplify with surds.
Worked examples
WE 1
Basic GDC use — given limits
Find the area between the curve y = x² + 1, the x-axis, and the lines x = 0 and x = 3.
step 1 — set upA = ∫(0 to 3) (x² + 1) dxstep 2 — GDCType into GDC: ∫ from 0 to 3 of (x² + 1)GDC output: 12A = 12 square unitswhen the answer is whole, no conversion needed!
WE 2
Decimal output → convert to exact
The region R is bounded by y = x⁴ − 2x² + 5, the x-axis, x = 1, and x = 2. Find the exact area, in the form p/q.
step 1 — set upA = ∫(1 to 2) (x⁴ − 2x² + 5) dxstep 2 — GDCGDC output: 6.5333333…step 3 — convert to exactUse GDC’s “decimal → fraction” function:6.5333… = 98/15A = 98/15 square unitsdecimals lose marks when “exact” is asked. always convert!
WE 3
Limits from where the curve crosses the x-axis
Find the area enclosed by the curve y = 4 − x² and the x-axis.
step 1 — find the limitsSolve 4 − x² = 0 → x² = 4 → x = ±2Limits: a = −2, b = 2step 2 — set upA = ∫(−2 to 2) (4 − x²) dxstep 3 — GDCGDC: 32/3A = 32/3 square unitsno vertical lines given → roots of f(x) = 0 are the limits!
WE 4
Area under a trig curve
Find the area between y = sin x, the x-axis, x = 0 and x = π.
setupA = ∫(0 to π) sin x dxGDCMake sure GDC is in RADIANS!GDC output: 2A = 2 square unitsalways check radians mode for trig integrals!
WE 5
Area under an exponential curve
Find the area enclosed by y = e^x, the x-axis, x = 0 and x = 2. Give your answer in the form a + be^c.
setupA = ∫(0 to 2) e^x dxGDCGDC: 6.389056… or e² − 1exact forme² − 1 = −1 + e², so a = −1, b = 1, c = 2A = e² − 1 square unitswhen “in form a + be^c” appears, look for an exact e-form on your GDC!
💡 Top tips
Sketch first — use your GDC’s graphing screen to see the area you want.
Find the limits before integrating — either from vertical lines or from f(x) = 0.
Always convert decimals to exact form if the question asks. Use your GDC’s fraction or simplify function.
Radians mode for any trig integral.
No “+ c“ needed for definite integrals — they cancel.
Practise with your specific GDC before the exam — every model has slightly different menus.
⚠ Common mistakes
Leaving a decimal answer when the question asks for exact form (e.g. “in the form p/q”).
Wrong limits — using a y-value or mixing up which is upper/lower.
Adding “+ c“ to a definite integral. Not needed.
GDC in degrees for a trig integral. Always check radians.
Skipping the sketch — leads to wrong limits or wrong region.
🎉 You’ve finished the Integration series! You can now find antiderivatives, pin down constants from given points, and compute exact areas using a GDC. Calculus toolkit complete.
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