Last Updated Sep 27, 2025

Area Between Two Curves Simplified Revision Notes for Leaving Cert Mathematics

Revision notes with simplified explanations to understand Area Between Two Curves quickly and effectively.

Learn about Area Under/Between Curves for your Leaving Cert Mathematics Exam. This Revision Note includes a summary of Area Under/Between Curves for easy recall in your Mathematics exam

328+ students studying

Area Under/Between Curves Quizzes

Test your knowledge with quizzes.

Area Under/Between Curves Flashcards

Practice with bite-sized questions.

Area Under/Between Curves Questions by Topic

Prepare with real exam question.

Print Free PDF

Area Between Two Curves

Integration: Further Area

Area Between Two Curves

Find the area of the shaded region $A$.

Area between the uppermost curve and the $x$-axis, take the area between the lowermost curve and the $x$-axis.

$$A_1 = \int_0^1 x^2 \, dx = \left[ \frac{1}{3}x^3 \right]_0^1 = \left( \frac{1}{3}(1)^3 - \frac{1}{3}(0)^3 \right) = :highlight[\frac{1}{3}]$$ $$A_2 = \int_0^1 x^3 \, dx = \left[ \frac{1}{4}x^4 \right]_0^1 = \left( \frac{1}{4}(1)^4 - \frac{1}{4}(0)^4 \right) = :highlight[\frac{1}{4}]$$

Therefore:

$$A = \frac{1}{3} - \frac{1}{4} = :success[\frac{1}{12}]$$

Alternative Method

$$\int_0^1 x^2 - x^3 \, dx = \left[ \frac{1}{3}x^3 - \frac{1}{4}x^4 \right]_0^1 = \left( \frac{1}{3}(1)^3 - \frac{1}{4}(1)^4 \right) - \left( \frac{1}{3}(0)^3 - \frac{1}{4}(0)^4 \right) = :success[\frac{1}{12}]$$

---

infoNote

Q1

The diagram shows parts of the curves $y = x^2 + 1$ and $y = 11 - \frac{9}{x^2}$, which intersect at (1, 2) and (3, 10). Use integration to find the exact area of the shaded region enclosed between the two curves.

1. Find the area between the "upper curve" (red) and the x-axis:

$$\int_1^3 \left( 11 - \frac{9}{x^2} \right) dx = \int_1^3 11 - 9x^{-2} dx$$ $$= \left[ 11x + 9{x^{-1}} \right]_1^3$$ $$= (11(3) + 9(3)^{-1}) - (11(1) + 9(1)^{-1}) = :highlight[16]$$

2. Find the area between the "lower curve" (blue) and the $x$-axis:

$$\int_1^3 \left( x^2 + 1 \right) dx = \left[ \frac{x^3}{3} + x \right]_1^3 = \left( \frac{3^3}{3} + 3 \right) - \left( \frac{1^3}{3} + 1 \right) = :highlight[\frac{32}{3}]$$

3. Subtract:

$$A = 16 - \frac{32}{3} = :success[\frac{16}{3}]$$

---

Alternative Method: Subtract Before Evaluating Limits

$$\int_1^3 11 - 9{x^{-2}} - (x^2 + 1)\ dx = \int_1^3 10 - 9{x^{-2}} - x^2 \ dx$$ $$= \left[ 10x + 9{x^{-1}} - \frac{x^3}{3} \right]_1^3$$ $$= \left( 10(3) + 9(3)^{-1} - \frac{3^3}{3} \right) - \left( 10(1) + 9(1)^{-1} - \frac{1^3}{3} \right) = :success[\frac{16}{3}]$$

---

Only available for registered users.

Sign up now to view the full note, or log in if you already have an account!

Sign up with GoogleSign up with Apple

500K+ Students Use These Powerful Tools to Master Area Between Two Curves For their Leaving Cert Exams.

Enhance your understanding with flashcards, quizzes, and exams—designed to help you grasp key concepts, reinforce learning, and master any topic with confidence!

Other Revision Notes related to Area Between Two Curves you should explore

Discover More Revision Notes Related to Area Between Two Curves to Deepen Your Understanding and Improve Your Mastery

96%

114 rated

Area Under/Between Curves

Area Under Curve and x-axis

357+ studying

195KViews

View Note

96%

114 rated

Area Under/Between Curves

Area Between Curve and y-axis

270+ studying

191KViews

View Note

96%

114 rated

Area Under/Between Curves

Area Under Curve and x-axis

293+ studying

183KViews

View Note

96%

114 rated

Area Under/Between Curves

Area Between Curve and y-axis

333+ studying

192KViews

View Note