What is the general formula for the area between two curves y = f(x) and y = g(x) from x = a to x = b?

\(\int_a^b \big(f(x) - g(x)\big) \, dx\), where f(x) is the upper curve and g(x) is the lower curve. The order matters: always subtract the lower from the upper.

How do you find the limits of integration when finding the area between two curves?

The limits are the x-coordinates of the points where the curves intersect. Set \(f(x) = g(x)\) and solve for x. These intersection points bound the enclosed region.

What must you do when two curves intersect and cross over each other?

Split the area into separate integrals. At each intersection, the roles of \"upper\" and \"lower\" swap. Integrate (upper \(-\) lower) in each sub-interval, then add the results.

Why is it important to identify which curve is upper and which is lower?

If you subtract in the wrong order, you get a negative result. The area between curves must be positive, so upper minus lower ensures the integrand is positive in the region.

What prerequisites are needed before studying area between two curves?

Area under a curve (definite integration) and area between a curve and a line. You need to understand how to find areas using integration and how to handle intersections.

What is the integrand when finding the area between y = f(x) and y = g(x)?

\(f(x) - g(x)\) or \(g(x) - f(x)\), depending on which is upper. The integrand is always (upper curve) minus (lower curve).

What happens if you forget to split the integral when curves cross?

You would integrate (upper − lower) over the whole interval, but the upper/lower roles swap at the intersection. The result would be wrong — you would subtract area that should be added.

When might the area between two curves need to be split into more than one integral?

When the curves intersect and cross over. In each sub-interval between intersections, a different curve may be on top, so you need separate integrals for each part.

What is the relationship between area between two curves and area under a curve?

Area between two curves = area under upper curve minus area under lower curve. So \(\int (f - g) \, dx = \int f \, dx - \int g \, dx\) over the same limits.

How can a calculator help when finding area between two curves?

Use it to solve \(f(x) = g(x)\) for intersections (especially cubic equations) and to evaluate definite integrals numerically.

Library Podcasts

Courses

Referral & Rewards

Area Between 2 Curves

Summary

The area between two curves is found by integrating the difference between the upper curve and the lower curve with respect to x. When curves intersect and cross over, the region must be split into separate integrals. This topic builds on area under a curve and area between a curve and a line.

1. Definition and Prerequisites

The area between two curves is the region enclosed by two graphs between their points of intersection.

Ensure you are familiar with: area under a curve (definite integration of (y = f(x))) and area between a curve and a line.

In general, find the definite integral of "upper curve" minus "lower curve" between the intersection points.

The area depends on which region is being found and whether the curves intersect and cross over.

2. General Method

The area between two curves (y = f(x)) (upper) and (y = g(x)) (lower) from (x = a) to (x = b) is: [\int_a^b \big(f(x) - g(x)\big) , dx]

If the curves cross, identify which is upper and which is lower in each sub-interval.

The area may have to be split into separate integrals when the curves intersect.

The points at which curves intersect may need to be calculated by solving (f(x) = g(x)).

Diagram showing area R between two curves y=f(x) and y=g(x) from x=a to x=b

3. Step-by-Step Procedure

STEP 1: Find the intersections of the curves if needed. Solve (f(x) = g(x)) to get the limits.

STEP 2: Form the integral using the intersections as limits. Use "upper curve" (-) "lower curve" and evaluate the definite integral.

STEP 3: Repeat STEP 2 if more than one area is needed (e.g. when curves cross).

STEP 4: Add the areas together to find the total enclosed area.

4. Curves That Cross Over

5. Examiner Tips

Area Between 2 Curves

Summary

1. Definition and Prerequisites

The area between two curves is the region enclosed by two graphs between their points of intersection.

Ensure you are familiar with: area under a curve (definite integration of (y = f(x))) and area between a curve and a line.

In general, find the definite integral of "upper curve" minus "lower curve" between the intersection points.

The area depends on which region is being found and whether the curves intersect and cross over.

2. General Method

The area between two curves (y = f(x)) (upper) and (y = g(x)) (lower) from (x = a) to (x = b) is: [\int_a^b \big(f(x) - g(x)\big) , dx]

If the curves cross, identify which is upper and which is lower in each sub-interval.

The area may have to be split into separate integrals when the curves intersect.

The points at which curves intersect may need to be calculated by solving (f(x) = g(x)).

Diagram showing area R between two curves y=f(x) and y=g(x) from x=a to x=b

3. Step-by-Step Procedure

STEP 1: Find the intersections of the curves if needed. Solve (f(x) = g(x)) to get the limits.

STEP 2: Form the integral using the intersections as limits. Use "upper curve" (-) "lower curve" and evaluate the definite integral.

STEP 3: Repeat STEP 2 if more than one area is needed (e.g. when curves cross).

STEP 4: Add the areas together to find the total enclosed area.

4. Curves That Cross Over

When two curves intersect and cross, the "upper" and "lower" roles swap at the intersection.

Split the integral at each intersection point. In each sub-interval, identify which curve is on top.

For each sub-interval, integrate (upper (-) lower) and take the modulus if needed for signed area.

Sum the absolute values of each integral to get the total area.

5. Examiner Tips

If no diagram is provided, sketch one, even if the curves are not accurate.

Add information to any given diagram as you work: mark intersections, limits, and shade the required area.

Maximise use of your calculator: solving equations (especially cubic) and finding definite integrals.

Always verify which curve is upper and which is lower before subtracting.