IB Maths AA SL Topic 5 — Calculus Paper 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 = ∫_a^b 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 = ∫_a^b 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

Find the integral function. Look for the ∫□_□^□ button — physical or in a menu.

Enter the integrand f(x) — the function to integrate.

Enter the lower limit a and upper limit b.

Press enter. Read the answer — usually a decimal.

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 up A = ∫(0 to 3) (x² + 1) dxstep 2 — GDC Type into GDC: ∫ from 0 to 3 of (x² + 1) GDC output: 12 A = 12 square units when 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 up A = ∫(1 to 2) (x⁴ − 2x² + 5) dxstep 2 — GDC GDC output: 6.5333333…step 3 — convert to exact Use GDC’s “decimal → fraction” function: 6.5333… = 98/15 A = 98/15 square units decimals 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 limits Solve 4 − x² = 0 → x² = 4 → x = ±2 Limits: a = −2, b = 2step 2 — set up A = ∫(−2 to 2) (4 − x²) dxstep 3 — GDC GDC: 32/3 A = 32/3 square units no 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.

setup A = ∫(0 to π) sin x dxGDC Make sure GDC is in RADIANS! GDC output: 2 A = 2 square units always 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.

setup A = ∫(0 to 2) e^x dxGDC GDC: 6.389056… or e² − 1exact form e² − 1 = −1 + e², so a = −1, b = 1, c = 2 A = e² − 1 square units when “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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